tomleslie

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These are answers submitted by tomleslie

in the past, is usually associated with a bootleg/cracked copy of Maple

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  1. Most non-lineear ODEs don't have  analyic solutions, despite the impression given by your textbook
  2. The solution of any non-linear ODE (actually an linear ODE as well) is highly dependent on the values of any parameters and initial boundary conditions. Your exaqmaple provides no initial conditions (so I made them up) and no parameter values (other than ranges), so I made these up as well. The attached shows a couple of very different "solutions", depending on exactly which values you pick for parameters/ICs

ode := 0 = diff(y(x), x) + ((r + 2*x)*(p - y(x)^(-s)))/(-(b*y(x) - x^2)*s*y(x)^(-s - 1));

#parameters(0 < r, 0 < p, b < 1 and 0 < b, 0 < s)
sol:=dsolve([ode,y(0)=0.1], numeric, parameters=[ r, p, b, s],maxfun=0);

0 = diff(y(x), x)-(r+2*x)*(p-y(x)^(-s))/((b*y(x)-x^2)*s*y(x)^(-s-1))

 

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := [r = r, p = p, b = b, s = s]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 1, (2) = 1, (3) = 0, (4) = 0, (5) = 4, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 0, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .0, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..5, {(1) = .1, (2) = Float(undefined), (3) = Float(undefined), (4) = Float(undefined), (5) = Float(undefined)})), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..1, {(1) = .1}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8]), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..1, {(1, 1) = .0, (2, 0) = .0, (2, 1) = .0, (3, 0) = .0, (3, 1) = .0, (4, 0) = .0, (4, 1) = .0, (5, 0) = .0, (5, 1) = .0, (6, 0) = .0, (6, 1) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x)]`; if Y[1] < 0 then YP[1] := undefined; return 0 end if; if Y[1] < 0 then YP[1] := undefined; return 0 end if; YP[1] := (Y[2]+2*X)*(Y[3]-Y[1]^(-Y[5]))/((-X^2+Y[1]*Y[4])*Y[5]*Y[1]^(-Y[5]-1)); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x)]`; if Y[1] < 0 then YP[1] := undefined; return 0 end if; if Y[1] < 0 then YP[1] := undefined; return 0 end if; YP[1] := (Y[2]+2*X)*(Y[3]-Y[1]^(-Y[5]))/((-X^2+Y[1]*Y[4])*Y[5]*Y[1]^(-Y[5]-1)); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..5, {(1) = 0., (2) = .1, (3) = undefined, (4) = undefined, (5) = undefined}); _vmap := array( 1 .. 1, [( 1 ) = (1)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [x, y(x)], (4) = [r = r, p = p, b = b, s = s]}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(1)

sol(parameters=[0.1, 2.0, 10.0, 0.3]):
plots:-odeplot( sol, [x,y(x)], x=0..10 );

 

sol(parameters=[0.1, 2.0, 10.0, 0.4]):
plots:-odeplot( sol, [x,y(x)], x=0..10 );

Warning, cannot evaluate the solution further right of .64031848, probably a singularity

 

 

 

Download nonlinODE.mw

 

Exactly what is supposed to be "wrong" with this worksheet

The atached shows your worksheet running in Maple 18. The only problems arise becuase of the inclusion of the commands

set 1;

at various points. These errors do not surprise me, since this command obeys no Maple syntax and is in fact completely meaningless. Other than your use of meaningless commands, what else is supposed to be wrong with the attached??

restart:
interface(version);

`Standard Worksheet Interface, Maple 18.02, Windows 7, October 20 2014 Build ID 991181`

(1)

restart;
P := -lambda*exp(-Phi(xi))-mu*exp(Phi(xi));
u[0] := A[0]+A[1]*exp(-Phi(xi))+A[2]*exp(-Phi(xi))*exp(-Phi(xi));
u[1] := diff(u[0], xi);
d[1] := -A[1]*P*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*P;
d[2] := -A[1]*(lambda*P*exp(-Phi(xi))-mu*P*exp(Phi(xi)))*exp(-Phi(xi))+A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*P*exp(-Phi(xi))+4*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*P-2*A[2]*(exp(-Phi(xi)))^2*(lambda*P*exp(-Phi(xi))-mu*P*exp(Phi(xi)));
collect(expand((2*k*k)*w*beta*d[2]-(2*alpha*k*k)*d[1]-2*w*u[0]+k*u[0]*u[0]), exp(Phi(xi)));

-lambda*exp(-Phi(xi))-mu*exp(Phi(xi))

 

A[0]+A[1]*exp(-Phi(xi))+A[2]*(exp(-Phi(xi)))^2

 

-A[1]*(diff(Phi(xi), xi))*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*(diff(Phi(xi), xi))

 

-A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(-Phi(xi))-2*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))

 

-A[1]*(lambda*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(-Phi(xi))-mu*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(Phi(xi)))*exp(-Phi(xi))+A[1]*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))^2*exp(-Phi(xi))+4*A[2]*(exp(-Phi(xi)))^2*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))^2-2*A[2]*(exp(-Phi(xi)))^2*(lambda*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(-Phi(xi))-mu*(-lambda*exp(-Phi(xi))-mu*exp(Phi(xi)))*exp(Phi(xi)))

 

4*k^2*w*beta*A[2]*mu^2-2*alpha*k^2*A[1]*mu+k*A[0]^2-2*w*A[0]+(4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1])/exp(Phi(xi))+(16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2])/(exp(Phi(xi)))^2+(4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2])/(exp(Phi(xi)))^3+(12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2)/(exp(Phi(xi)))^4

(2)

restart;
solve({12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2, 4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2], 4*beta*k^2*mu^2*w*A[2]-2*alpha*k^2*mu*A[1]+k*A[0]^2-2*w*A[0], 4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1], 16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2]}, {k, w, A[0], A[1], A[2]});

set 1;

{k = 0, w = 0, A[0] = A[0], A[1] = A[1], A[2] = A[2]}, {k = k, w = w, A[0] = 0, A[1] = 0, A[2] = 0}, {k = k, w = w, A[0] = 2*w/k, A[1] = 0, A[2] = 0}, {k = RootOf(24*beta*lambda*mu*_Z^2-1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (1/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2-1)), A[1] = (1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2-1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2-1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}, {k = RootOf(24*beta*lambda*mu*_Z^2+1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (3/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2+1)), A[1] = -(1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2+1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2+1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}

 

Error, unexpected number

 

restart;
solve({12*beta*k^2*lambda^2*w*A[2]+k*A[2]^2, 4*beta*k^2*lambda^2*w*A[1]-4*alpha*k^2*lambda*A[2]+2*k*A[1]*A[2], 4*beta*k^2*mu^2*w*A[2]-2*alpha*k^2*mu*A[1]+k*A[0]^2-2*w*A[0], 4*beta*k^2*lambda*mu*w*A[1]-4*alpha*k^2*mu*A[2]+2*k*A[0]*A[1]-2*w*A[1], 16*beta*k^2*lambda*mu*w*A[2]-2*alpha*k^2*lambda*A[1]+2*k*A[0]*A[2]+k*A[1]^2-2*w*A[2]}, {k, w, A[0], A[1], A[2]});

set 1;

{k = 0, w = 0, A[0] = A[0], A[1] = A[1], A[2] = A[2]}, {k = k, w = w, A[0] = 0, A[1] = 0, A[2] = 0}, {k = k, w = w, A[0] = 2*w/k, A[1] = 0, A[2] = 0}, {k = RootOf(24*beta*lambda*mu*_Z^2-1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (1/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2-1)), A[1] = (1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2-1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2-1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}, {k = RootOf(24*beta*lambda*mu*_Z^2+1), w = RootOf(100*mu*lambda*_Z^2+1)*alpha/beta, A[0] = (3/2)*RootOf(100*mu*lambda*_Z^2+1)*alpha/(beta*RootOf(24*beta*lambda*mu*_Z^2+1)), A[1] = -(1/10)*alpha/(beta*mu*RootOf(24*beta*lambda*mu*_Z^2+1)), A[2] = -12*RootOf(24*beta*lambda*mu*_Z^2+1)*lambda^2*RootOf(100*mu*lambda*_Z^2+1)*alpha}

 

Error, unexpected number

 

restart;
solve({24*Z^2*beta*lambda*mu-1}, {Z});
solve({100*Z^2*lambda*mu+1}, {Z});

{Z = (1/12)*6^(1/2)/(beta*lambda*mu)^(1/2)}, {Z = -(1/12)*6^(1/2)/(beta*lambda*mu)^(1/2)}

 

{Z = -(1/10)/(-lambda*mu)^(1/2)}, {Z = (1/10)/(-lambda*mu)^(1/2)}

(3)

restart;
k := (1/12)*sqrt(6)/sqrt(beta*lambda*mu);
w := -alpha/((10*sqrt(-lambda*mu))*beta);
A[0] := 1/2*(-alpha/((10*sqrt(-lambda*mu))*((1/12)*beta*sqrt(6)/sqrt(beta*lambda*mu))));
A[1] := (1/10)*alpha/((1/12)*beta*mu*sqrt(6)/sqrt(beta*lambda*mu));
A[2] := (12*(1/12))*sqrt(6)*lambda^2*alpha/(sqrt(beta*lambda*mu)*(10*sqrt(-lambda*mu)));
lambda := 3;
mu := 2;
H := -ln(sqrt(lambda/mu)*tan(sqrt(lambda*mu)*(xi+C)));
u[0] := A[0]+A[1]*exp(-H)+A[2]*exp(-H)*exp(-H);
f := diff(u[0], xi);
S := diff(f, xi);
simplify(%);

(1/12)*6^(1/2)/(beta*lambda*mu)^(1/2)

 

-(1/10)*alpha/((-lambda*mu)^(1/2)*beta)

 

-(1/10)*alpha*6^(1/2)*(beta*lambda*mu)^(1/2)/((-lambda*mu)^(1/2)*beta)

 

(1/5)*alpha*6^(1/2)*(beta*lambda*mu)^(1/2)/(beta*mu)

 

(1/10)*6^(1/2)*lambda^2*alpha/((beta*lambda*mu)^(1/2)*(-lambda*mu)^(1/2))

 

3

 

2

 

-ln((1/2)*6^(1/2)*tan(6^(1/2)*(xi+C)))

 

(1/10)*alpha*(-6)^(1/2)/beta^(1/2)+(3/10)*alpha*6^(1/2)*tan(6^(1/2)*(xi+C))/beta^(1/2)-(9/40)*alpha*(-6)^(1/2)*tan(6^(1/2)*(xi+C))^2/beta^(1/2)

 

(9/5)*alpha*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)-(9/20)*alpha*(-6)^(1/2)*tan(6^(1/2)*(xi+C))*6^(1/2)*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)

 

(18/5)*alpha*tan(6^(1/2)*(xi+C))*6^(1/2)*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)-(27/10)*alpha*(-6)^(1/2)*(1+tan(6^(1/2)*(xi+C))^2)^2/beta^(1/2)-(27/5)*alpha*(-6)^(1/2)*tan(6^(1/2)*(xi+C))^2*(1+tan(6^(1/2)*(xi+C))^2)/beta^(1/2)

 

(9/10)*alpha*6^(1/2)*((6*I)*cos(6^(1/2)*(xi+C))^2+4*sin(6^(1/2)*(xi+C))*cos(6^(1/2)*(xi+C))-9*I)/(cos(6^(1/2)*(xi+C))^4*beta^(1/2))

(4)

 

Download satisfy18.mw

 

are shown in the attached

  d1:= [[3, 11], [4, 6], [5, 8]]:
#
# The 'quick' way - direct calculatiom
#
  add(i[1]-2, i in d1);

6

(1)

#
# If you really want a procedure/function -
# probably only worth it if the process is
# being applied to several input lists.
#
# Define the function
#
  f:=lis-> local i:
           add( i[1]-2, i in lis):
#
# Apply the function to the argument
#
  f(d1);

6

(2)

 

Download addCoord.mw

In the attached, I have made a few adjustments

  1. I don't have access to Maple's GlobalOptimization  add-on - so this code section is commented out. Although the best answer achieved (ie 6.979) should be borne in mind for future reference.
  2. I rewrote the code around the DirectSearch() section - just so that it would run, there were various syntax issues! The result isn't pretty but it works
    1. Using the GlobalOptima() command from the DirectSearch package produce a "maximum" of 3.95463824386087 - so obviously not as "maximal" as the value obtained in (1) above
    2. Rather than using DirectSearch:-GlobalOptima(), I tried defining 20 different initial points (you can change this number via N__inits), just to see how well a basic DirectSearch:-Search() command would do, running from different start points. This procuced a "maximum" of 4.17769689355030. So better than that obtained form DirectSearch:-GlobalOptima, but not as good as the GlobalOptimization add-on
  3. I think this variable behaviour really just illustrates the whole issue of optimization for non-convex functions. There is no known algorithm which is guaranteed to find the optimal solution. Nearly all of the approaches I am familiar with perform the following steps (for maximization)
    1. Guess an initial point
    2. Do a few evaluations of the objective function around this point to figure out which way is "uphill"
    3. Step "uphill"
    4. Repeat steps (2)-(3) untial nothing is "uphill" any more - this is a "local" maximum
    5. Guess other initial points and repeat (2)-(3) above, retaining that which provides the highest "local" maximum
    6. Make the (somewhat rash) statement, that the "highest" of all the "local" maxima obtained is in fact the "global" maxima
    7. The success/failure of this strategy is almost completely governed by the number and selection of the initial points. Covering a ssufficient(?) number of initial points depends on the number of variables in the objecive function and what sort of "spacing/range" of intial points makes sense. The latter criterion is an "art" not a "science"
  4. MUch more interesting (to me at least) is the final execution group in the original worksheet, which by a "coordinate transformation" changes the problem to a linear program which can be solve quickly, and provides the same solution as Maple's GlobalOptimization add-on. This is impressive, and I'm still trying to get my head around why it works. If I were you, this is where I would focus my attention

Anyhow, for what it is worth, and bearing in mind all of the caveats above, see the attached

Portfolio Optimization with the Omega Ratio

 

Introduction

 

 

Traditional investment performance benchmarks, like the Sharpe Ratio, approximate the returns distribution with mean and standard deviation. This, however, assumes the distribution is normal.  Many modern investments vehicles, like hedge funds, display fat tails, and skew and kurtosis in the returns distribution. Hence, they cannot be adequately benchmarked with traditional approaches.

 

One solution, proposed by Shadwick and Keating in 2002 is the Omega Ratio.  This divides the returns distribution into two halves – the area below a target return, and the above a target return. The Omega Ratio is simply the former divided by the latter. A higher value is better.

 

For a set of discrete returns, the Omega Ratio is given by

 

Omega(L) = E[max(R-L, 0)]/E[max(L-R, 0)]

 

where L is a target return and R is a vector of returns.

 

This application finds the asset weights that maximize the Omega Ratio of a portfolio of ten investments, given their simulated monthly returns and a target return.

 

This is a non-convex problem, and requires global optimizers for a rigorous solution. However, a transformation of the variables (only valid for Omega Ratios of over 1) converts the optimization into a linear program.

 

This application implements both approaches, the former using Maple's Global Optimization Toolbox, and the latter using Maple's linear programming features. For the data set provided in this application, both approaches give comparible results.

 

Returns Data and Minimum Acceptable Return

 

restart

Monthly hedge fund returns

Number of funds

N := LinearAlgebra[ColumnDimension](data)

10

(2.1)

Number of returns for each fund

S := LinearAlgebra[RowDimension](data)

36

(2.2)

Target Return

L := .1

Omega Ratio

 

OmegaRatio := proc (L, returns, weights) local weightedReturns, above, below, N, S, a, i, j; if convert(`~`[type](weights, numeric), set)[] then N := LinearAlgebra[ColumnDimension](returns); S := LinearAlgebra[RowDimension](returns); weightedReturns := [seq(add(returns[i, j]*weights[j], j = 1 .. N), i = 1 .. S)]; below := select(proc (x) options operator, arrow; evalb(x <= L) end proc, weightedReturns); above := select(proc (x) options operator, arrow; evalb(L < x) end proc, weightedReturns); return add(a-L, `in`(a, above))/add(L-a, `in`(a, below)) else return 'procname(args)' end if end proc

 

"Strawman" portfolio of equal weights at the target return

OmegaRatio(L, data, [.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])

HFloat(4.003192510007608)

(2.3)

Global Optimization

 

NULL

 

Optimized Omega Ratio

NULL

Optimized investment weights

NULL

Direct Search

 

with(DirectSearch); wrapper := proc () global L, data; OmegaRatio(L, data, [args]) end proc; ansGlobal := GlobalOptima(wrapper(seq(w[i], i = 1 .. 10)), [add(w[i], i = 1 .. N) = 1, seq(w[i] = 0 .. .1, i = 1 .. N)], maximize); r := rand(0. .. 1.0); N__inits := 20; for j to N__inits do rvals := seq(r(), j = 1 .. N); ans[j] := Search(wrapper(seq(w[i], i = 1 .. N)), [add(w[i], i = 1 .. N) = 1, seq(w[i] = 0 .. .1, i = 1 .. N)], initialpoint = [seq(w[i] = `~`[`/`]([rvals], add(rvals))[i], i = 1 .. N)], usewarning = false, maximize) end do; seq(ans[j][1], j = 1 .. N__inits); ans[max[index]([seq(ans[j][1], j = 1 .. N__inits)])]

[HFloat(3.954638243860872), [w[1] = HFloat(0.1), w[2] = HFloat(0.1), w[3] = HFloat(0.09714257075732306), w[4] = HFloat(0.09710635877067376), w[5] = HFloat(0.09968784671681309), w[6] = HFloat(0.09353148613677527), w[7] = HFloat(0.09999450830780217), w[8] = HFloat(0.09999999999998163), w[9] = HFloat(0.0806020956977236), w[10] = HFloat(0.09959361372477742)], 1780]

 

HFloat(3.865771689021921), HFloat(4.023426840990234), HFloat(3.7192361069502007), HFloat(3.995249533082313), HFloat(3.9377227987097942), HFloat(3.929660158669612), HFloat(3.976521807439104), HFloat(3.8353678058307317), HFloat(4.177696893550296), HFloat(3.905839821607468), HFloat(3.8932377942132868), HFloat(3.955000171243558), HFloat(3.8678107038369087), HFloat(3.7031708987131884), HFloat(3.867738407138106), HFloat(3.879604472834744), HFloat(3.989758201454413), HFloat(3.7912897819280573), HFloat(4.016230388564598), HFloat(3.9414606304823896)

 

[HFloat(4.177696893550296), [w[1] = HFloat(0.09565526251705606), w[2] = HFloat(0.09996201616302475), w[3] = HFloat(0.045971685565773905), w[4] = HFloat(0.09896111064147), w[5] = HFloat(0.09797375100631699), w[6] = HFloat(0.09999918334598228), w[7] = HFloat(0.1), w[8] = HFloat(0.07669946978292375), w[9] = HFloat(0.1), w[10] = HFloat(0.1)], 2276]

(2.3.1)

Linear Program

 

 

The transformation of the optimization problem into a linear program is described here

eq1 := seq(add(data[i, j]*w[j], j = 1 .. N)-u[i]+d[i]-L*t = 0, i = 1 .. S)

eq2 := add(w[j], j = 1 .. N) = t

eq3 := add(d[i]/S, i = 1 .. S) = 1

obj := add(u[i]/S, i = 1 .. S)

cons := seq([u[i] >= 0, d[i] >= 0][], i = 1 .. S), seq([w[j] >= 0][], j = 1 .. N)

resultsLP := Optimization[LPSolve](obj, {cons, eq1, eq2, eq3}, maximize, assume = nonnegative)


Optimized Omega Ratio

resultsLP[1]

6.97971754550025

(2.4.1)

Optimized investment weights

assign(select(has, resultsLP[2], t))

weightsLP := map(proc (i) options operator, arrow; lhs(i) = rhs(i)/t end proc, select(has, resultsLP[2], w))

[w[1] = HFloat(0.0), w[2] = HFloat(0.0), w[3] = HFloat(0.0), w[4] = HFloat(0.40979022216167177), w[5] = HFloat(0.0), w[6] = HFloat(0.21961738132814781), w[7] = HFloat(0.33321118343926637), w[8] = HFloat(0.0), w[9] = HFloat(0.0), w[10] = HFloat(0.037381213348692446)]

(2.4.2)

 

 

Download optProb.mw

 

 

 

see the attached

  restart:
  with(plots):
  with(Student[Precalculus]):
  with(ColorTools):
#
# Define some parameters
#
  p__I:=[0.2,0.2]:
  p__B:=0.8:
  eps:=0.4:
  pfm__B:=1:
  pfm__A:=1:
#
# Equation of line
#
  l1:=Line(p__I, [1,1])[1]:

#
# Define regions
#
  nreg:= 7:
  x__b:= rhs(isolate(l1,x)):
  y__b:= max(p__B, rhs(l1)):
  regions:= [ [ x>=p__I[1], x<=eps,    y>=p__B,    y<=pfm__B  ],
              [ x>=eps,     x<=pfm__A, y>=y__b,    y<=pfm__B  ],
              [ x>=x__b,    x<=pfm__A, y>=p__B,    y<=pfm__B  ],
              [ x>=p__I[1], x<=eps,    y>=rhs(l1), y<=p__B    ],
              [ x>=p__I[1], x<=eps,    y>=p__I[2], y<=rhs(l1) ],
              [ x>=eps,     x<=x__b,   y>=rhs(l1), y<=p__B    ],
              [ x>=eps,     x<=pfm__A, y>=p__I[2], y<=y__b    ]
            ]:
  pltopts:= x=0..1, y=0..1, nolines:
  display( [seq( inequal( regions[j],  pltopts, color=COLOR(HUE, j/nreg)), j=1..nreg)])

 

 


 

Download ineqPlt.mw

then only general guidelines can be given. But maybe you want something like the attached

  restart:

  with(plots):
  animate( pointplot3d,
           [ [cos(t),sin(t),t],
             symbol=solidsphere,
             symbolsize=50, color=blue
           ],
           t=0..4*Pi,
           background = spacecurve
                        ( [cos(t), sin(t), t],
                          t=0..4*Pi,
                          color =red,
                          thickness=2
                        ),
           frames=100
         );

 

 

Download anim.mw

 

 

Unfortunately the integrand contains the "unknown" variables 'v' and 'u'. If I give these numerical values then a (numerical) answer can be obtained. See the attached

 

restart;
Q:=3:
a:=10:
c:=5:
b:=3:
d:=0:
psi:=Int(Int(exp(-a*(mu-b)^2)*exp(-c*(V-d)^2)*exp(I*(Q*u-(5/3*Q)*v^(mu/V)-3/16*(v^(4/V)/Q)))*v^((1/2)*(1-V)/V), mu = 0 .. 2), V = -15 .. 15);
evalf(%);
v:=1:
u:=1:
evalf(psi);

Int(Int(exp(-10*(mu-3)^2)*exp(-5*V^2)*exp(I*(3*u-5*v^(mu/V)-(1/16)*v^(4/V)))*v^((1/2)*(1-V)/V), mu = 0 .. 2), V = -15 .. 15)

 

Int(Int(exp(-10.*(mu-3.)^2)*exp(-5.*V^2)*exp((1.*I)*(3.*u-5.*v^(mu/V)-0.6250000000e-1*v^(4./V)))*v^(.5000000000*(1.-1.*V)/V), mu = 0. .. 2.), V = -15. .. 15.)

 

-0.8122177805e-6-0.1516524210e-5*I

(1)

 


Download intNum.mw

The heat equation in a uniform rod depends on both position and time - nothing in your post indicates a time-dependence, so I don't really understand the question.

Can only recommend that you enter

?examples,pdsolve_boundaryconditions

at the Maple prompt. This worksheet contains the heat equation with various different boundary conditions amongst the examples

Since you are have a fairly simple piecewise curve, fitting splines seems a bit "excessive".

However the final execution group in the attached shows how to do it.

restart;

#Digits:=500:

with(CurveFitting):
with(plots):
with(plottools):

f := x -> piecewise(0 < x and x <= 5000, 0, 5000 < x and x <= 7500, 0.008*x - 40, 7500 < x and x <= 10000, -0.008*x + 80, 10000 < x and x <= 15000, 0);

f := proc (x) options operator, arrow; piecewise(0 < x and x <= 5000, 0, 5000 < x and x <= 7500, 0.8e-2*x-40, 7500 < x and x <= 10000, (-1)*0.8e-2*x+80, 10000 < x and x <= 15000, 0) end proc

(1)

Ics:= plot(f(x), x = 0 .. 15000,color = blue, legend = ["Initial Condition"],labels = ["x", "concentration"],axis = [tickmarks = [5, subticks = 1]]);

 

g := t -> piecewise(0 < t and t <= 1, 0, 1 < t and t <= 3, 15*t - 15, 3 < t and t <= 6, -10*t + 60, 6 < t and t <= 10, 0);

g := proc (t) options operator, arrow; piecewise(0 < t and t <= 1, 0, 1 < t and t <= 3, 15*t-15, 3 < t and t <= 6, -10*t+60, 6 < t and t <= 10, 0) end proc

(2)

Bc:= plot(g(t), t = 0 .. 10,color = green, legend = ["Boundary Condition"],labels = ["time(hr)", "concentration"],axis = [tickmarks = [10, subticks = 1]]);

 

NULL

NULL

v := .5

alpha := 15

mu := v/(2*alpha)

beta := v^2/(4*alpha)

NULL

"h(x):=(f(x))/((e)^(-mu*x)):"

IC := plot(h(x), x = 0 .. 15000)

 

NULL

"r(t):=(g(t))/((e)^(-beta*t)):"

BC := plot(r(t), t = 0 .. 10)

 

#
# Gnerate a Spline fit to data from the above curve
# but why??????? - only OP knows
#
# then plot the resulting fit
#
  A:= unapply
      ( Spline
        ( getdata( BC)[3][..,1],
          getdata( BC)[3][..,2],
          'z',
          degree=1
        ),
        z
      ):
  plot(A, 0..10);
#
# Generate a few values from the fitted curve, just to
# show that everything is working "more or less" properly
#
 A(1); A(2); A(3); A(4); A(5);A(6); A(7):
#
# For comparison
#
  r(1); r(2); r(3); r(4); r(5); r(6);
#
# So I suppose the "fit" isn't bad, but if I just wanted
# a Matrix of values for the function r(t)
#
  mat:= Matrix([seq( [j, r(j)], j=0..8,0.1)]);

 

0.122174450291075e-1

 

15.1255479294070

 

30.3459526079177

 

20.3361031946834

 

10.2104962632121

 

0.288592862691956e-1

 

0.

 

15.12552228

 

30.37735355

 

20.33612661

 

10.21051862

 

0.

 

_rtable[18446744074370485894]

(3)

 


 

Download splineFit.mw

both in your  command

pduA := pdsolve({pdeu,pdev}, {IBCu union IBCv}, numeric, time = t, range = -0.1e-02 .. 2, spacestep = 1/65, timestep = .1);

  1. IBCu union IBCv will return a set: so {IBCu union IBCv} will return a set containing  a set - which is syntactically incorrect
  2. You have boundary conditions specified at x=0: you cannnot define option range = -0.1e-02 .. 2, sinc x=0 is no longer a boundary

Hence if I change the above command to

pduA := pdsolve({pdeu,pdev}, IBCu union IBCv, numeric, time = t, range = 0 .. 2, spacestep = 1/65, timestep = .1);

it returns the usual solution module. See the attached
 

Two 1-D coupled classical Burgers equations

We advance the above case to account for u, v interaction but still in the abscence of the quantum potential terms in the quantum momentum rate equations.  This results in a coupled pair of classical Burgers equations, which have applications in hydrodynamic suspensions, plasma physics, nonlinear optics.  For example, S.E. Esipov (Phys. Rev. E, 52 (1995), pp. 3711-3718) [preprint: https://arxiv.org/pdf/cond-mat/9501021.pdf] is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids, under the effect of gravity. Thus, for that example, u is the main fluid, while v is the suspension.

u, v

 

u

 

v

(1)

hBar := 'hBar': m := 'm':Fu := 'Fu': Fv := 'Fv': # define constants

m := 1:Fu := 0.2:Fv := 0.1: # set constant values (hBar := 1: not used)

pdeu := diff(u(x,t),t)+u(x,t)/m*(diff(u(x,t),x)) = Fu;

diff(u(x, t), t)+u(x, t)*(diff(u(x, t), x)) = .2

(2)

pdev := diff(v(x,t),t)+u(x,t)/m*(diff(v(x,t),x)) = Fv;

diff(v(x, t), t)+u(x, t)*(diff(v(x, t), x)) = .1

(3)

By inspection of the derivatives in above equations we set up the ICs and BCs for u(x,t) and v(x,t):

IBCu := {u(x,0) = 0.1*sin(2*Pi*x),u(0,t) = 0.5-0.5*cos(2*Pi*t)};# IBC for u

{u(0, t) = .5-.5*cos(2*Pi*t), u(x, 0) = .1*sin(2*Pi*x)}

(4)

IBCv := {v(x,0) = 0.2*sin((1/2)*Pi*x),v(0,t)=0.2-0.2*cos(2*Pi*t)};# IBC for v

{v(0, t) = .2-.2*cos(2*Pi*t), v(x, 0) = .2*sin((1/2)*Pi*x)}

(5)

IBC := IBCu union IBCv;

{u(0, t) = .5-.5*cos(2*Pi*t), u(x, 0) = .1*sin(2*Pi*x), v(0, t) = .2-.2*cos(2*Pi*t), v(x, 0) = .2*sin((1/2)*Pi*x)}

(6)

We set up the equations and boundary conditions to fully specify the equations and initial and boundary conditions (IBC) to obtain a 'numerical solution' for u(x,t) and v(x,t):

pds := pdsolve([pdeu,pdev], IBC, time = t, range = 0..2,numeric);# numerical solution

_m700556416

(7)

Now specify the 3D plot of the solutions u(x,t),v(x,t), setting transparency =" 0.5" so that both solution surfaces are visible. (Note that the plot instruction is labelled by the solution prefix pds:- and its suffix is plot3d.)

Error, incorrect syntax in parse: `.` unexpected (near 1st character of parsed string)

" 0.5"

 

T := 30; p1 := pds:-plot3d({[u, color = red], [v, color = cyan]}, t = 0 .. T, x = -0.1e-2 .. 2,transparency = 0.5, numpoints = 1000, color = red, orientation = [-146, 54, 0], title = print("Coupled solution \n u(x, t) red,v(x,t) blue", numeric));

 

 

30

 

"Coupled solution 
 u(x, t) red,v(x,t) blue", numeric

 

 

Here is a 1-D plot at time T = 30:

T := 30; p1 := pds:-plot({[u,color=red],[v,color=blue]},t = T,numpoints = 1000,color=[red,blue], gridlines = true, title = print("u(x, t) red,v(x,t) blue, numeric"));

30

 

"u(x, t) red,v(x,t) blue, numeric"

 

 

The coupled Re and Im momenta have common features. Recall that u, v are subject to forces  0.2 , 0.1  respectively; hence the difference in their momentum levels. [As a check we should obtain solutions to the    Schrdiff(o(t), t, t)dinger equation and then derive the momentum beables (u,v) from the wavefunction.]

We now implement an animation of the solution:

pduA := pdsolve({pdeu,pdev}, IBCu union IBCv, numeric, time = t, range = 0 .. 2, spacestep = 1/65, timestep = .1);

_m875144000

(8)

``

NULL


 

Download pdeProb.mw

 

the attached is what you want

If not you are going to have to explain your issue more clearly
 

restart

with(plots)

with(CurveFitting)

Digits := 2

with(LinearAlgebra)

g := x -> piecewise(0 < x and x <= 10, x, 10 < x and x <= 20, -x+20);

g := proc (x) options operator, arrow; piecewise(0 < x and x <= 10, x, 10 < x and x <= 20, -x+20) end proc

(1)

p0 := plot(g(x), x = 0 .. 20)

 

plottools:-getdata(p0); M := %[-1]; ExportMatrix("E:/p0.xls", M)

["curve", [0. .. 20., 0. .. 9.99982415678391945], _rtable[18446744074378701510]]

 

_rtable[18446744074378701510]

(2)

v := .6

alpha := 15

mu := v/(2*alpha)

beta := v^2/(4*alpha)

NULL

#
# The following plots x-M*exp(-mu*x) for x=1.
# Fot other values of 'x' change the final option
# to x=2,2,-1, whatever
#
  plots:-pointplot( eval~(x-M*~exp(-mu*x), x=1));

 

``

Download plotMAt.mw

Achieving the attached convinced me that all calculations should be done "unitless" - and (only if strictly necessary) should units be applied at the end!

I'm not even sure that it is "complete" becuase the units of the expression returned by dsolve() ought to be 'm', but the expression *appears* to be "unitless"

Anyhow, for what it is worth

restart;
with(Units):
assume(t>0);

Automatically loading the Units[Simple] subpackage

 

 

T := 60*Unit('s');
mass := 80*Unit('kg');
b__1 := 13*Unit('kg')/Unit('s');
g := -9.81*Unit('m')/Unit('s')/Unit('s');
b := piecewise( 0 <= t*Unit('s') and t*Unit('s') < T,
                b__1,
                T<= t*Unit('s') and t*Unit('s') < infinity*Unit('s'),
                7*b__1
              );

Acceleration := diff(h(t)*Unit('m'), t*Unit('s'), t*Unit('s')) = g - b*diff(h(t)*Unit('m'), t*Unit('s'))/mass;

IC := h(0)*Unit('m') = 4000*Unit('m'),
      D(h)(0)*Unit(('m')/('s')) = 0.1*Unit('m'/'s');

solution := dsolve({IC, Acceleration}, h(t));

T := 60*Unit('s')

 

mass := 80*Unit('kg')

 

b__1 := 13*Unit('kg'/'s')

 

g := -9.81*Unit('m'/'s'^2)

 

b := piecewise(t < 60, 13, 60 <= t, 91)*Unit('kg'/'s')

 

Acceleration := diff(h(t), t, t) = -9.81-(1/80)*piecewise(t < 60, 13, 60 <= t, 91)*(diff(h(t), t))

 

IC := h(0) = 4000, (D(h))(0) = .1

 

h(t) = piecewise(t < 60, -(3924/65)*t-(62888/169)*exp(-(13/80)*t)+738888/169, 60 <= t, -(3924/455)*t+10118760/8281-(8984/169)*exp(-(91/80)*t+117/2)-(53904/169)*exp(-39/4)+(376704/8281)*exp(-(91/80)*t+273/4))

(1)

fsolve( rhs(solution), t);
plot( rhs(solution),
      t = 0 .. 150,
      y = 0 .. 4100,
      axis[2] = [gridlines = 30],
      axis[1] = [gridlines = 20],
      labels = ["t [s]", "h(t) [m]"],
      labelfont = ["Times", 15],
      color = "red", size = [1000, 1000]
    )

141.6838339

 

 

 


 

Download unitProb.mw

as in the attached. On this site, the animation appears to be runniing at 10 frames/sec (the default rate), but if you download the worksheet, you can use the worksheet animation toolbar to

  1. set the update rate to 1 frame/sec
  2. decide whether you wnat the animation to play a singel cycle or continuously
  3. manuallly step frame-by-frame

  restart;
  rnge:=-1..1:
  fcns:=[x, x^2, 1-x]:
  plots:-display( [ seq
                    ( plot
                      ( fcns[j], x=rnge),
                      j=1..3
                    )
                  ],
                  insequence=true
                );

 

 

 

 

 

Download slowanim.mw

One can use brute force, because it is relatively trivial to code, then check to see how bad timings get as the number of odd inputs and even addends is increased.

On my machine, the brute force approach in the attached produced the following approximate timings

#    Nodd    Neven      real time
#
#    100     100          15  msecs
#    100     1000        15  msecs
#    100     10000      15  msecs
#   
#    1000    100          66  msecs
#    1000    1000        77  msecs
#    1000    10000      1.0 secs
#  
#    10000   100         0.5 secs
#    10000   1000       1.5 secs
#    10000   10000      16  secs     

Out of idle curiosity, with Nodd=10000 and Neven =10000, there were 4534 "successful" odd numbers and 463 "failures".

A brief examination of the "failures" up to these limits indicates that they always have 3 as a prime factor - coincidence???

NB It would not surprise me if there were a much more efficient way to implement this test

  restart;
#
# Define the upper limit for the odd numbers
# to be tested and the upper limit for the
# even addends
#
  Nodd:=10000:
  Neven:=10000:
#
# Procedure which does the work
#
  doWork:= proc( p::posint, N::posint );
                 local f:=NumberTheory:-PrimeFactors(p),
                       j:
                 for j from 2 by 2 to N do
                     if   isprime~(f+~j)={true}
                     then #
                          # Return a list containing the
                          # input odd number, its prime
                          # factors, the lowest even number
                          # which produces a a new set of primes,
                          # and this new set of primes
                          #
                            return [p, f, j, f+~j];
                     fi;
                 od;
                 return #
                        # No even addend found: return a list
                        # containing the input odd number and
                        # a zero
                        #
                          [p, f, 0];
           end proc:

#
# Check how long it takes to return the complete list
# of answers for the specific values of NO and NE
#
# Results on my machine were. NB all timings approximate
#
#    Nodd    Neven      real time
#
#    100     100        15  msecs
#    100     1000       15  msecs
#    100     10000      15  msecs
#   
#    1000    100        66  msecs
#    1000    1000       77  msecs
#    1000    10000      1.0 secs
#  
#    10000   100        0.5 secs
#    10000   1000       1.5 secs
#    10000   10000      16  secs     
#
  ans:=CodeTools:-Usage([seq( doWork(j, Neven), j=3..Nodd,2)]):

memory used=1.73GiB, alloc change=40.00MiB, cpu time=14.98s, real time=15.04s, gc time=577.20ms

 

#
# Check answers for a couple of the cases used by OP
# to illustrate problem
#
# Utility to facilitate lookup of the answer in the
# main result list for any supplied odd number
#
  getVal:=p->(p-1)/2:
  ans[ getVal(119) ];
  ans[ getVal(105) ];

[119, {7, 17}, 6, {13, 23}]

 

[105, {3, 5, 7}, 0]

(1)

#
# Split the original list into successes and failures.
# Out of idle curiosity, check the number of successes
# and failures
#
  success,failure:=selectremove(i->numelems(i)=4, ans):
  numelems(success);
  numelems(failure);
#
# Output the first few successes and failures
#
  success[1..100];
  failure[1..100];
#
# Check whether the failure cases *always* have '3' as
# a prime factor. This should output any odd number
# which "failed" to generate a new set of prime numbers,
# and itself doesn't have '3' as a prime factor.
#
# It outputs nothing  interesting? coincidence??
#
  seq( `if`( j[2][1]=3, NULL, j[1]), j in failure);

4534

 

465

 

[[3, {3}, 2, {5}], [5, {5}, 2, {7}], [7, {7}, 4, {11}], [9, {3}, 2, {5}], [11, {11}, 2, {13}], [13, {13}, 4, {17}], [15, {3, 5}, 2, {5, 7}], [17, {17}, 2, {19}], [19, {19}, 4, {23}], [21, {3, 7}, 4, {7, 11}], [23, {23}, 6, {29}], [25, {5}, 2, {7}], [27, {3}, 2, {5}], [29, {29}, 2, {31}], [31, {31}, 6, {37}], [33, {3, 11}, 2, {5, 13}], [35, {5, 7}, 6, {11, 13}], [37, {37}, 4, {41}], [39, {3, 13}, 4, {7, 17}], [41, {41}, 2, {43}], [43, {43}, 4, {47}], [45, {3, 5}, 2, {5, 7}], [47, {47}, 6, {53}], [49, {7}, 4, {11}], [51, {3, 17}, 2, {5, 19}], [53, {53}, 6, {59}], [55, {5, 11}, 2, {7, 13}], [57, {3, 19}, 4, {7, 23}], [59, {59}, 2, {61}], [61, {61}, 6, {67}], [63, {3, 7}, 4, {7, 11}], [65, {5, 13}, 6, {11, 19}], [67, {67}, 4, {71}], [69, {3, 23}, 8, {11, 31}], [71, {71}, 2, {73}], [73, {73}, 6, {79}], [75, {3, 5}, 2, {5, 7}], [77, {7, 11}, 6, {13, 17}], [79, {79}, 4, {83}], [81, {3}, 2, {5}], [83, {83}, 6, {89}], [85, {5, 17}, 2, {7, 19}], [87, {3, 29}, 2, {5, 31}], [89, {89}, 8, {97}], [91, {7, 13}, 4, {11, 17}], [93, {3, 31}, 10, {13, 41}], [95, {5, 19}, 12, {17, 31}], [97, {97}, 4, {101}], [99, {3, 11}, 2, {5, 13}], [101, {101}, 2, {103}], [103, {103}, 4, {107}], [107, {107}, 2, {109}], [109, {109}, 4, {113}], [111, {3, 37}, 4, {7, 41}], [113, {113}, 14, {127}], [115, {5, 23}, 6, {11, 29}], [117, {3, 13}, 4, {7, 17}], [119, {7, 17}, 6, {13, 23}], [121, {11}, 2, {13}], [123, {3, 41}, 2, {5, 43}], [125, {5}, 2, {7}], [127, {127}, 4, {131}], [129, {3, 43}, 4, {7, 47}], [131, {131}, 6, {137}], [133, {7, 19}, 4, {11, 23}], [135, {3, 5}, 2, {5, 7}], [137, {137}, 2, {139}], [139, {139}, 10, {149}], [141, {3, 47}, 14, {17, 61}], [143, {11, 13}, 6, {17, 19}], [145, {5, 29}, 2, {7, 31}], [147, {3, 7}, 4, {7, 11}], [149, {149}, 2, {151}], [151, {151}, 6, {157}], [153, {3, 17}, 2, {5, 19}], [155, {5, 31}, 6, {11, 37}], [157, {157}, 6, {163}], [159, {3, 53}, 8, {11, 61}], [161, {7, 23}, 6, {13, 29}], [163, {163}, 4, {167}], [165, {3, 5, 11}, 2, {5, 7, 13}], [167, {167}, 6, {173}], [169, {13}, 4, {17}], [171, {3, 19}, 4, {7, 23}], [173, {173}, 6, {179}], [175, {5, 7}, 6, {11, 13}], [177, {3, 59}, 2, {5, 61}], [179, {179}, 2, {181}], [181, {181}, 10, {191}], [183, {3, 61}, 10, {13, 71}], [185, {5, 37}, 6, {11, 43}], [187, {11, 17}, 2, {13, 19}], [189, {3, 7}, 4, {7, 11}], [191, {191}, 2, {193}], [193, {193}, 4, {197}], [197, {197}, 2, {199}], [199, {199}, 12, {211}], [201, {3, 67}, 4, {7, 71}], [203, {7, 29}, 12, {19, 41}], [205, {5, 41}, 2, {7, 43}]]

 

[[105, {3, 5, 7}, 0], [195, {3, 5, 13}, 0], [231, {3, 7, 11}, 0], [285, {3, 5, 19}, 0], [315, {3, 5, 7}, 0], [357, {3, 7, 17}, 0], [429, {3, 11, 13}, 0], [465, {3, 5, 31}, 0], [483, {3, 7, 23}, 0], [525, {3, 5, 7}, 0], [555, {3, 5, 37}, 0], [585, {3, 5, 13}, 0], [609, {3, 7, 29}, 0], [627, {3, 11, 19}, 0], [645, {3, 5, 43}, 0], [663, {3, 13, 17}, 0], [693, {3, 7, 11}, 0], [735, {3, 5, 7}, 0], [855, {3, 5, 19}, 0], [861, {3, 7, 41}, 0], [897, {3, 13, 23}, 0], [915, {3, 5, 61}, 0], [945, {3, 5, 7}, 0], [969, {3, 17, 19}, 0], [975, {3, 5, 13}, 0], [987, {3, 7, 47}, 0], [1005, {3, 5, 67}, 0], [1023, {3, 11, 31}, 0], [1071, {3, 7, 17}, 0], [1095, {3, 5, 73}, 0], [1113, {3, 7, 53}, 0], [1131, {3, 13, 29}, 0], [1155, {3, 5, 7, 11}, 0], [1185, {3, 5, 79}, 0], [1221, {3, 11, 37}, 0], [1239, {3, 7, 59}, 0], [1287, {3, 11, 13}, 0], [1311, {3, 19, 23}, 0], [1365, {3, 5, 7, 13}, 0], [1395, {3, 5, 31}, 0], [1419, {3, 11, 43}, 0], [1425, {3, 5, 19}, 0], [1449, {3, 7, 23}, 0], [1455, {3, 5, 97}, 0], [1491, {3, 7, 71}, 0], [1545, {3, 5, 103}, 0], [1575, {3, 5, 7}, 0], [1581, {3, 17, 31}, 0], [1599, {3, 13, 41}, 0], [1617, {3, 7, 11}, 0], [1635, {3, 5, 109}, 0], [1653, {3, 19, 29}, 0], [1665, {3, 5, 37}, 0], [1743, {3, 7, 83}, 0], [1755, {3, 5, 13}, 0], [1785, {3, 5, 7, 17}, 0], [1827, {3, 7, 29}, 0], [1833, {3, 13, 47}, 0], [1869, {3, 7, 89}, 0], [1881, {3, 11, 19}, 0], [1887, {3, 17, 37}, 0], [1905, {3, 5, 127}, 0], [1935, {3, 5, 43}, 0], [1989, {3, 13, 17}, 0], [1995, {3, 5, 7, 19}, 0], [2013, {3, 11, 61}, 0], [2067, {3, 13, 53}, 0], [2079, {3, 7, 11}, 0], [2085, {3, 5, 139}, 0], [2121, {3, 7, 101}, 0], [2139, {3, 23, 31}, 0], [2145, {3, 5, 11, 13}, 0], [2193, {3, 17, 43}, 0], [2205, {3, 5, 7}, 0], [2211, {3, 11, 67}, 0], [2247, {3, 7, 107}, 0], [2265, {3, 5, 151}, 0], [2301, {3, 13, 59}, 0], [2325, {3, 5, 31}, 0], [2337, {3, 19, 41}, 0], [2355, {3, 5, 157}, 0], [2373, {3, 7, 113}, 0], [2409, {3, 11, 73}, 0], [2415, {3, 5, 7, 23}, 0], [2445, {3, 5, 163}, 0], [2499, {3, 7, 17}, 0], [2535, {3, 5, 13}, 0], [2541, {3, 7, 11}, 0], [2553, {3, 23, 37}, 0], [2565, {3, 5, 19}, 0], [2583, {3, 7, 41}, 0], [2607, {3, 11, 79}, 0], [2625, {3, 5, 7}, 0], [2679, {3, 19, 47}, 0], [2691, {3, 13, 23}, 0], [2697, {3, 29, 31}, 0], [2715, {3, 5, 181}, 0], [2745, {3, 5, 61}, 0], [2751, {3, 7, 131}, 0], [2769, {3, 13, 71}, 0]]

(2)

 

Download prmeProb.mw

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