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About numerical parameters...

bogo 55
ode differential-equation parameter
April 06 2016
0 2

I am solving a complicated ODE and I would like to know if there is a way for Maple to output the ODEs without doing any numerical substitutions for known parameters. Say one of my parameters, call it P, is initialized (there are many more but Ill just simplify and consider one here) to a value of 10. In order to chekc that I have coded the ODEs correctly it would help me if Maple does not substitue with the numerical value 10 for P when displayng teh ODEs, but rather keeps P, as a parameter. Is there a way to achieve this?

Function defined as integral in ODE...

bogo 55 Maple 16
ode eval
July 29 2015
2 18

I have a function defined as an integral that can be computed numerically at every point, and MAPLE does it. When I use this function inside an ODE for another function, maple complains that unknown parameters must be initialized. It sees the integration variable in my definition of the function defined as an integral as a parameter. Is there a work around?

Attached you will find an worksheet with my example. The function defined as an integral is called YFD(x) and the independent function I am trying to evaluate numerically using dsolve is Y(x).

 

 

restart

g := 2

2

 (1)

## The function  that is defined as an integral and will enter the ODE

YFD := proc (x) options operator, arrow; evalf((1/2)*g*(Int((xi^2-x^2)^(1/2)*xi/(exp(xi)+1), xi = x .. infinity))/Pi^2) end proc

proc (x) options operator, arrow; evalf((1/2)*g*(Int((xi^2-x^2)^(1/2)*xi/(exp(xi)+1), xi = x .. infinity))/Pi^2) end proc

 (2)

YFD(2)

0.9953007436e-1

 (3)

ode := diff(Y(x), x)+10^5*(Y(x)^2-YFD(x)^2)/x^2 = 0

diff(Y(x), x)+100000*(Y(x)^2-0.1026598225e-1*(Int((xi^2-1.*x^2)^(1/2)*xi/(exp(xi)+1.), xi = x .. Float(infinity)))^2)/x^2 = 0

 (4)

IC := Y(1) = YFD(1)

Y(1) = .1534968659

 (5)

Attempt to solve ODE:

 

sol := dsolve({IC, ode}, numeric, range = 1 .. 100)

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 := [xi = xi]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 19, [( 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..53, {(1) = 1, (2) = 1, (3) = 0, (4) = 0, (5) = 1, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 1, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 30000, (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) = 0, (52) = 0, (53) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = 1.0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = 1.0, (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..2, {(1) = .1534968659, (2) = 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..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..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..2, {(1) = .0, (2) = .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 = float[8], order = C_order)]), ( 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[2]^2-1.*X^2 < 0 then YP[1] := undefined; return 0 end if; YP[1] := -100000*(Y[1]^2-0.1026598225e-1*(Int(evalf((Y[2]^2-1.*X^2)^(1/2))*Y[2]/(exp(Y[2])+1.), Y[2] = X .. Float(infinity)))^2)/X^2; 0 end proc, -1, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([1, table( [( 1, "name" ) = xi, ( 1, "local" ) = xi, ( 1, "value" ) = undefined ] )]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = Y(x)]`; if Y[2]^2-1.*X^2 < 0 then YP[1] := undefined; return 0 end if; YP[1] := -100000*(Y[1]^2-0.1026598225e-1*(Int(evalf((Y[2]^2-1.*X^2)^(1/2))*Y[2]/(exp(Y[2])+1.), Y[2] = X .. Float(infinity)))^2)/X^2; 0 end proc, -1, 0, 0, 0, 0]) ] )) ] ); _y0 := Array(0..2, {(1) = 1., (2) = .1534968659}); _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); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `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; 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] < 10 and 10 <= _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 10 <= _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] < 10 and 10 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 10 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-10 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-10; 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] < 10 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 < 1 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 _src = 0 and 10 < _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 <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> 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 10 < _dat[4][9] then if _dat[4][9]-10 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _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]-10, 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 <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> 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 10 < _dat[4][9] then if _dat[4][9]-10 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-10 = _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]-10, 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]; _dat[4][26] := _EnvDSNumericSaveDigits; _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) = [xi = xi]}); _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; _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

 (6)

sol(1)

[x = HFloat(1.0), Y(x) = .1534968659]

 (7)

sol(3)

Error, (in sol) parameter 'xi' must be assigned a numeric value before obtaining a solution

  

``

 

Download DMProdMaplePrimes.mw

dsolve method=lsode and range option...

bogo 55 Maple
dsolve numeric
April 09 2015
0 2

Is it possible to use an option similar to range when using lsode method for dsolve? The ODEs I am trying to solve is stiff and will not work with the flag stiff=true or with method=rosenbrock unless i set Digits:=20. I want to avoid doing that as much as possible, since I believe wit will be very taxing, computationally. I have a very large systeom to solve. I found that method=lsode works with the default Digits=15. 

 

However I need to have the solutions in a given range stored for future access and manipulations. Using range gives me an error: 

Error, (in dsolve/numeric/an_args/lsode) lsode keyword was range, optional keyword must be one of 'ctrl', 'initial', 'itask', 'output', 'procedure', 'procvars', 'start', 'number', 'abserr', 'relerr', 'maxfun', 'minstep', 'maxstep', 'initstep', 'startinit', 'implicit', 'optimize', 'complex'

 

I cannot figure out how to use range or something similar with lsode. Anyone knows? 

Large system of ODEs ...

bogo 55
dsolve numeric
April 09 2015
0 3

Hi! I have to solve a very large system of ODEs for a set of functions that can be indexed with two integers, n and l, call them f[n,l](x), where n ranges from 1 to an order 1000 number and l ranges from 1 to an order 10 number. 

What is the best strategy in dsolve, keeping in mind I will only need to have quick access to the values of the first few such functions, say f[0,0](x) and f[0,1](x) for any given x in a given range. I dont need to store the rest, and trying to store all of them actually will give me a warning about length of output exceeds limit of 1000000. So, having maple output the result of dsolve in a procedural form seems to be what I want. However in doing so, for evey value of x i call the solving procedure maple will solve the system each time, thus taking a long time. 

 

My dilema is: to save storage space and memory I want to use dsolve output as procedures. To save time, when accessing the functions I need (only the first few in the large set of unknown functions that obey the ODE) for a given value of x, I would like maple to have those already stored instead of computing them with each invocation. So, what I need is a somehow split behavior of output. For some of my unknown functions to act as if i use range in dsolve, so that I have access at the values without further computation, and for the rest of the solutions (most of the unknown functions) just keep them in procedural form. How do I achieve this? 

 

Below is a schematic of my probem:

 

ODEs:={Set of about 10000 coupled ODEs for functions labeled f[n,l] n ranging from 1..1000 and l from 1..10}

 

dsolve(ODEs,numeric, output=???)#What options to put in dsolve such that I have quick access to f[0,0](x) and f[0,1](x) in a given range (say xmin=0.1 and xmax=10 for example) but not store the rest. 

 

 

Combinations of numerical solutions from...

bogo 55 Maple
dsolve stiff
April 07 2015
0 3

Hi! I am trying to plot and store in memory some specific combinations of the solutions of the systems of ODEs that I get numerically from dsolve for a particular range of the independent variable. 

A particular case for my problem is the following system of stiff ODEs for two unknown functions f[0,0](x) and f[1,0](x) beween xini (where the Initial conditions are defined) and xfin, an arbitrary value of x. Note that rosebrock method does not work, and I can only solve it with lsode[adamsfull] or lsode[backfull]. I am attaching a maple file that shows what I have done.


``

############## System of ODEs that needs to be solved ####################################

xini := .1

.1

 (1)

xfin := 2

2

 (2)

SystemToSolve := diff(f[0, 0](x), x)+(2./x^5+.5000000000/x)*f[0, 0](x) = -15.58845727*sin(.5773502693*x)/x^2+(46.76537182*(3.*sin(.5773502693*x)-1.732050808*x*cos(.5773502693*x)))/x^4, diff(f[1, 0](x), x)+(6./x^5+1.500000000/x)*f[1, 0](x)-1.*f[0, 0](x)/x = (-15.58845727*sin(.5773502693*x)/x^2+(46.76537182*(3.*sin(.5773502693*x)-1.732050808*x*cos(.5773502693*x)))/x^4)*(1.-1.*(1/x^4)^(1/4)*exp(1/x^4)*GAMMA(.7500000000, 1/x^4))-(1.*(-10.39230485*sin(.5773502693*x)/x^2+(31.17691454*(3.*sin(.5773502693*x)-1.732050808*x*cos(.5773502693*x)))/x^4+(4.*((.8660254040*(3.*sin(.5773502693*x)-1.732050808*x*cos(.5773502693*x)))/x+(.8660254040*((3.*(1.-6./x^2))*sin(.5773502693*x)+10.39230485*cos(.5773502693*x)/x))/x))/((1/x^4)^(1/4)*exp(1/x^4)*GAMMA(.7500000000, 1/x^4)*x^5)))*(1/x^4)^(1/4)*exp(1/x^4)*GAMMA(.7500000000, 1/x^4), f[0, 0](xini) = 1.503498546, f[1, 0](xini) = -.5011661819:

 

###################################################################################

 

``

ListProcs := dsolve({SystemToSolve}, numeric, method = lsode[backfull], output = listprocedure):

f00 := eval(f[0, 0](x), ListProcs);

proc (x) local _res, _dat, _solnproc, _xout, _ndsol, _pars, _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) else _xout := evalf(x) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _n, _y0, _ctl, _octl, _reinit, _errcd, _fcn, _i, _yini, _pars, _ini, _par; option `Copyright (c) 2002 by the University of Waterloo. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _ctl := array( 1 .. 34, [( 1 ) = (2), ( 2 ) = (.1), ( 3 ) = (.1), ( 4 ) = (1), ( 5 ) = (1), ( 6 ) = (22), ( 7 ) = (0), ( 9 ) = (-.5011661819), ( 8 ) = (1.503498546), ( 11 ) = (0.1e-6), ( 10 ) = (0.1e-6), ( 13 ) = (0), ( 12 ) = (0), ( 15 ) = (0), ( 14 ) = (0), ( 18 ) = (0), ( 19 ) = (0), ( 16 ) = (0), ( 17 ) = (0), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = (0), ( 21 ) = (0), ( 27 ) = (0), ( 26 ) = (0), ( 25 ) = (0), ( 24 ) = (0), ( 31 ) = (0), ( 30 ) = (0), ( 29 ) = (0), ( 28 ) = (0), ( 32 ) = (0), ( 33 ) = (-1), ( 34 ) = (0) ] ); _octl := array( 1 .. 34, [( 1 ) = (2), ( 2 ) = (.1), ( 3 ) = (.1), ( 4 ) = (1), ( 5 ) = (1), ( 6 ) = (22), ( 7 ) = (0), ( 9 ) = (-.5011661819), ( 8 ) = (1.503498546), ( 11 ) = (0.1e-6), ( 10 ) = (0.1e-6), ( 13 ) = (0), ( 12 ) = (0), ( 15 ) = (0), ( 14 ) = (0), ( 18 ) = (0), ( 19 ) = (0), ( 16 ) = (0), ( 17 ) = (0), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = (0), ( 21 ) = (0), ( 27 ) = (0), ( 26 ) = (0), ( 25 ) = (0), ( 24 ) = (0), ( 31 ) = (0), ( 30 ) = (0), ( 29 ) = (0), ( 28 ) = (0), ( 32 ) = (0), ( 33 ) = (-1), ( 34 ) = (0) ] ); _n := trunc(_ctl[1]); _yini := Array(0..2, {(1) = .1, (2) = 1.503498546}); _y0 := Array(0..2, {(1) = .1, (2) = 1.503498546}); _fcn := proc (N, X, Y, YP) option `[Y[1] = f[0,0](x), Y[2] = f[1,0](x)]`; YP[1] := -15.58845727*sin(.5773502693*X)/X^2+46.76537182*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X^4-(2./X^5+.5000000000/X)*Y[1]; if 1/X^4 < 0 then YP[1] := undefined; return 0 end if; if 1/X^4 < 0 then YP[1] := undefined; return 0 end if; YP[2] := (-15.58845727*sin(.5773502693*X)/X^2+46.76537182*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X^4)*(1.-1.*evalf((1/X^4)^(1/4))*exp(1/X^4)*GAMMA(.7500000000, 1/X^4))-1.*(-10.39230485*sin(.5773502693*X)/X^2+31.17691454*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X^4+4.*(.8660254040*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X+.8660254040*(3.*(1.-6./X^2)*sin(.5773502693*X)+10.39230485*cos(.5773502693*X)/X)/X)*evalf(1/(1/X^4)^(1/4))/(exp(1/X^4)*GAMMA(.7500000000, 1/X^4)*X^5))*evalf((1/X^4)^(1/4))*exp(1/X^4)*GAMMA(.7500000000, 1/X^4)-(6./X^5+1.500000000/X)*Y[2]+1.*Y[1]/X; 0 end proc; _pars := []; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then return _y0[0] elif _xout = "method" then return "lsode" elif _xout = "numfun" then return trunc(_ctl[24+trunc(_ctl[1])]) elif _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _ctl[2]-_y0[0] = 0. then error "no information is available on last computed point" else _xout := _ctl[2] end if elif _xout = "enginedata" then return eval(_octl, 1) elif _xout = "function" then return eval(_fcn, 1) 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); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _yini) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n, _ini, _yini, _pars) end if; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; _octl[2] := _y0[0]; _octl[3] := _y0[0]; for _i to _n do _octl[_i+7] := _y0[_i] end do; for _i to nops(_pars) do _octl[2*_n+30+_i] := _y0[_n+_i] end do; for _i to 34 do _ctl[_i] := _octl[_i] end do; if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] else return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] end if else return "procname" end if end if; if _xout-_y0[0] = 0. then return [seq(_y0[_i], _i = 0 .. _n)] end if; _reinit := false; if _xin <> "last" then if 0 < 0 and `dsolve/numeric/checkglobals`(0, table( [ ] ), _pars, _n, _yini) then _reinit := true; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; for _i to _n do _octl[_i+7] := _y0[_i] end do; for _i to nops(_pars) do _octl[2*_n+30+_i] := _y0[_n+_i] end do end if; if _pars <> [] and select(type, {seq(_yini[_n+_i], _i = 1 .. nops(_pars))}, 'undefined') <> {} then error "parameters must be initialized before solution can be computed" end if end if; if not _reinit and _xout-_ctl[2] = 0 then [_ctl[2], seq(_ctl[_i], _i = 8 .. 7+_n)] else if sign(_xout-_ctl[2]) <> sign(_ctl[2]-_y0[0]) or abs(_xout-_y0[0]) < abs(_xout-_ctl[2]) or _reinit then for _i to 34 do _ctl[_i] := _octl[_i] end do end if; _ctl[3] := _xout; if Digits <= evalhf(Digits) then try _errcd := evalhf(`dsolve/numeric/lsode`(_fcn, var(_ctl))) catch: userinfo(2, `dsolve/debug`, print(`Exception in lsode:`, [lastexception])); if searchtext('evalhf', lastexception[2]) <> 0 or searchtext('real', lastexception[2]) <> 0 or searchtext('hardware', lastexception[2]) <> 0 then _errcd := `dsolve/numeric/lsode`(_fcn, _ctl) else error end if end try else _errcd := `dsolve/numeric/lsode`(_fcn, _ctl) end if; if _errcd < 0 then userinfo(2, {dsolve, `dsolve/lsode`}, `Last values returned:`); userinfo(2, {dsolve, `dsolve/lsode`}, ` t =`, _ctl[2]); _i := 8; userinfo(2, {dsolve, `dsolve/lsode`}, ` y =`, _ctl[_i]); for _i from _i+1 to 7+_n do userinfo(2, {dsolve, `dsolve/lsode`}, ` `, _ctl[_i]) end do; if _errcd+1. = 0. then if _ctl[14+trunc(_ctl[1])] <> 0 then error "an excessive amount of work was done, maxstep may be too small" else error "an excessive amount of work (greater than mxstep) was done" end if elif _errcd+2. = 0. then error "too much accuracy was requested for the machine being used" elif _errcd+3. = 0. then error "illegal input was detected" elif _errcd+4. = 0. then error "repeated error test failures on the attempted step" elif _errcd+5. = 0. then error "repeated convergence test failures on the attempted step" elif _errcd+6. = 0. then error "pure relative error control requested for a variable that has vanished" elif _errcd+7. = 0. then error "cannot evaluate the solution past %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_ctl[2]) else error "unknown error code returned from lsode %1", trunc(_errcd) end if end if; if _Env_smart_dsolve_numeric = true then if _y0[0] < _xout and procname("right") < _xout then procname("right") := _xout elif _xout < _y0[0] and _xout < procname("left") then procname("left") := _xout end if end if; [_xout, seq(_ctl[_i], _i = 8 .. 7+_n)] end if end proc, (2) = Array(1..3, {(1) = 18446744078356217278, (2) = 18446744078356217454, (3) = 18446744078356217630}), (3) = [x, f[0, 0](x), f[1, 0](x)], (4) = []}); _solnproc := _dat[1]; _pars := map(rhs, _dat[4]); if not type(_xout, 'numeric') then if member(x, ["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, '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, ["last", 'last', "initial", 'initial', NULL]) then _res := _solnproc(convert(x, 'string')); if type(_res, 'list') then return _res[2] else return NULL end if elif member(x, ["parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [_res[2], seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] 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), 'string') = rhs(x); if lhs(_xout) = "initial" then if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else _res := _solnproc("initial" = ["single", 2, rhs(_xout)]) end if elif not type(rhs(_xout), 'list') then error "initial and/or parameter values must be specified in a list" elif lhs(_xout) = "initial_and_parameters" and nops(rhs(_xout)) = nops(_pars)+1 then _res := _solnproc(lhs(_xout) = ["single", 2, op(rhs(_xout))]) else _res := _solnproc(_xout) end if; if lhs(_xout) = "initial" then return _res[2] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [_res[2], 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), 'string') = rhs(x)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _dat[3] end if; if procname <> unknown then return ('procname')(x) else _ndsol := `tools/gensym`("f[0,0](x)"); eval(FromInert(_Inert_FUNCTION(_Inert_NAME("assign"), _Inert_EXPSEQ(ToInert(_ndsol), _Inert_VERBATIM(pointto(_dat[2][2])))))); return FromInert(_Inert_FUNCTION(ToInert(_ndsol), _Inert_EXPSEQ(ToInert(x)))) end if end if; try _res := _solnproc(_xout); _res[2] catch: error end try end proc

 (3)

f10 := eval(f[1, 0](x), ListProcs);

proc (x) local _res, _dat, _solnproc, _xout, _ndsol, _pars, _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) else _xout := evalf(x) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _n, _y0, _ctl, _octl, _reinit, _errcd, _fcn, _i, _yini, _pars, _ini, _par; option `Copyright (c) 2002 by the University of Waterloo. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _ctl := array( 1 .. 34, [( 1 ) = (2), ( 2 ) = (.1), ( 3 ) = (.1), ( 4 ) = (1), ( 5 ) = (1), ( 6 ) = (22), ( 7 ) = (0), ( 9 ) = (-.5011661819), ( 8 ) = (1.503498546), ( 11 ) = (0.1e-6), ( 10 ) = (0.1e-6), ( 13 ) = (0), ( 12 ) = (0), ( 15 ) = (0), ( 14 ) = (0), ( 18 ) = (0), ( 19 ) = (0), ( 16 ) = (0), ( 17 ) = (0), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = (0), ( 21 ) = (0), ( 27 ) = (0), ( 26 ) = (0), ( 25 ) = (0), ( 24 ) = (0), ( 31 ) = (0), ( 30 ) = (0), ( 29 ) = (0), ( 28 ) = (0), ( 32 ) = (0), ( 33 ) = (-1), ( 34 ) = (0) ] ); _octl := array( 1 .. 34, [( 1 ) = (2), ( 2 ) = (.1), ( 3 ) = (.1), ( 4 ) = (1), ( 5 ) = (1), ( 6 ) = (22), ( 7 ) = (0), ( 9 ) = (-.5011661819), ( 8 ) = (1.503498546), ( 11 ) = (0.1e-6), ( 10 ) = (0.1e-6), ( 13 ) = (0), ( 12 ) = (0), ( 15 ) = (0), ( 14 ) = (0), ( 18 ) = (0), ( 19 ) = (0), ( 16 ) = (0), ( 17 ) = (0), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = (0), ( 21 ) = (0), ( 27 ) = (0), ( 26 ) = (0), ( 25 ) = (0), ( 24 ) = (0), ( 31 ) = (0), ( 30 ) = (0), ( 29 ) = (0), ( 28 ) = (0), ( 32 ) = (0), ( 33 ) = (-1), ( 34 ) = (0) ] ); _n := trunc(_ctl[1]); _yini := Array(0..2, {(1) = .1, (2) = 1.503498546}); _y0 := Array(0..2, {(1) = .1, (2) = 1.503498546}); _fcn := proc (N, X, Y, YP) option `[Y[1] = f[0,0](x), Y[2] = f[1,0](x)]`; YP[1] := -15.58845727*sin(.5773502693*X)/X^2+46.76537182*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X^4-(2./X^5+.5000000000/X)*Y[1]; if 1/X^4 < 0 then YP[1] := undefined; return 0 end if; if 1/X^4 < 0 then YP[1] := undefined; return 0 end if; YP[2] := (-15.58845727*sin(.5773502693*X)/X^2+46.76537182*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X^4)*(1.-1.*evalf((1/X^4)^(1/4))*exp(1/X^4)*GAMMA(.7500000000, 1/X^4))-1.*(-10.39230485*sin(.5773502693*X)/X^2+31.17691454*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X^4+4.*(.8660254040*(3.*sin(.5773502693*X)-1.732050808*X*cos(.5773502693*X))/X+.8660254040*(3.*(1.-6./X^2)*sin(.5773502693*X)+10.39230485*cos(.5773502693*X)/X)/X)*evalf(1/(1/X^4)^(1/4))/(exp(1/X^4)*GAMMA(.7500000000, 1/X^4)*X^5))*evalf((1/X^4)^(1/4))*exp(1/X^4)*GAMMA(.7500000000, 1/X^4)-(6./X^5+1.500000000/X)*Y[2]+1.*Y[1]/X; 0 end proc; _pars := []; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then return _y0[0] elif _xout = "method" then return "lsode" elif _xout = "numfun" then return trunc(_ctl[24+trunc(_ctl[1])]) elif _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _ctl[2]-_y0[0] = 0. then error "no information is available on last computed point" else _xout := _ctl[2] end if elif _xout = "enginedata" then return eval(_octl, 1) elif _xout = "function" then return eval(_fcn, 1) 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); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _yini) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n, _ini, _yini, _pars) end if; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; _octl[2] := _y0[0]; _octl[3] := _y0[0]; for _i to _n do _octl[_i+7] := _y0[_i] end do; for _i to nops(_pars) do _octl[2*_n+30+_i] := _y0[_n+_i] end do; for _i to 34 do _ctl[_i] := _octl[_i] end do; if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [seq(_yini[_i], _i = 0 .. _n)] elif _xout = "parameters" then return [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] else return [seq(_yini[_i], _i = 0 .. _n)], [seq(_yini[_n+_i], _i = 1 .. nops(_pars))] end if else return "procname" end if end if; if _xout-_y0[0] = 0. then return [seq(_y0[_i], _i = 0 .. _n)] end if; _reinit := false; if _xin <> "last" then if 0 < 0 and `dsolve/numeric/checkglobals`(0, table( [ ] ), _pars, _n, _yini) then _reinit := true; if _pars <> [] then _par := {seq(rhs(_pars[_i]) = _yini[_n+_i], _i = 1 .. nops(_pars))}; for _i from 0 to _n do _y0[_i] := subs(_par, _yini[_i]) end do; for _i from _n+1 to _n+nops(_pars) do _y0[_i] := _yini[_i] end do else for _i from 0 to _n do _y0[_i] := _yini[_i] end do end if; for _i to _n do _octl[_i+7] := _y0[_i] end do; for _i to nops(_pars) do _octl[2*_n+30+_i] := _y0[_n+_i] end do end if; if _pars <> [] and select(type, {seq(_yini[_n+_i], _i = 1 .. nops(_pars))}, 'undefined') <> {} then error "parameters must be initialized before solution can be computed" end if end if; if not _reinit and _xout-_ctl[2] = 0 then [_ctl[2], seq(_ctl[_i], _i = 8 .. 7+_n)] else if sign(_xout-_ctl[2]) <> sign(_ctl[2]-_y0[0]) or abs(_xout-_y0[0]) < abs(_xout-_ctl[2]) or _reinit then for _i to 34 do _ctl[_i] := _octl[_i] end do end if; _ctl[3] := _xout; if Digits <= evalhf(Digits) then try _errcd := evalhf(`dsolve/numeric/lsode`(_fcn, var(_ctl))) catch: userinfo(2, `dsolve/debug`, print(`Exception in lsode:`, [lastexception])); if searchtext('evalhf', lastexception[2]) <> 0 or searchtext('real', lastexception[2]) <> 0 or searchtext('hardware', lastexception[2]) <> 0 then _errcd := `dsolve/numeric/lsode`(_fcn, _ctl) else error end if end try else _errcd := `dsolve/numeric/lsode`(_fcn, _ctl) end if; if _errcd < 0 then userinfo(2, {dsolve, `dsolve/lsode`}, `Last values returned:`); userinfo(2, {dsolve, `dsolve/lsode`}, ` t =`, _ctl[2]); _i := 8; userinfo(2, {dsolve, `dsolve/lsode`}, ` y =`, _ctl[_i]); for _i from _i+1 to 7+_n do userinfo(2, {dsolve, `dsolve/lsode`}, ` `, _ctl[_i]) end do; if _errcd+1. = 0. then if _ctl[14+trunc(_ctl[1])] <> 0 then error "an excessive amount of work was done, maxstep may be too small" else error "an excessive amount of work (greater than mxstep) was done" end if elif _errcd+2. = 0. then error "too much accuracy was requested for the machine being used" elif _errcd+3. = 0. then error "illegal input was detected" elif _errcd+4. = 0. then error "repeated error test failures on the attempted step" elif _errcd+5. = 0. then error "repeated convergence test failures on the attempted step" elif _errcd+6. = 0. then error "pure relative error control requested for a variable that has vanished" elif _errcd+7. = 0. then error "cannot evaluate the solution past %1, maxfun limit exceeded (see <a href='http://www.maplesoft.com/support/help/search.aspx?term=dsolve,maxfun' target='_new'>?dsolve,maxfun</a> for details)", evalf[8](_ctl[2]) else error "unknown error code returned from lsode %1", trunc(_errcd) end if end if; if _Env_smart_dsolve_numeric = true then if _y0[0] < _xout and procname("right") < _xout then procname("right") := _xout elif _xout < _y0[0] and _xout < procname("left") then procname("left") := _xout end if end if; [_xout, seq(_ctl[_i], _i = 8 .. 7+_n)] end if end proc, (2) = Array(1..3, {(1) = 18446744078356217278, (2) = 18446744078356217454, (3) = 18446744078356217630}), (3) = [x, f[0, 0](x), f[1, 0](x)], (4) = []}); _solnproc := _dat[1]; _pars := map(rhs, _dat[4]); if not type(_xout, 'numeric') then if member(x, ["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, '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, ["last", 'last', "initial", 'initial', NULL]) then _res := _solnproc(convert(x, 'string')); if type(_res, 'list') then return _res[3] else return NULL end if elif member(x, ["parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [_res[3], seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] 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), 'string') = rhs(x); if lhs(_xout) = "initial" then if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else _res := _solnproc("initial" = ["single", 3, rhs(_xout)]) end if elif not type(rhs(_xout), 'list') then error "initial and/or parameter values must be specified in a list" elif lhs(_xout) = "initial_and_parameters" and nops(rhs(_xout)) = nops(_pars)+1 then _res := _solnproc(lhs(_xout) = ["single", 3, op(rhs(_xout))]) else _res := _solnproc(_xout) end if; if lhs(_xout) = "initial" then return _res[3] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [_res[3], 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), 'string') = rhs(x)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _dat[3] end if; if procname <> unknown then return ('procname')(x) else _ndsol := `tools/gensym`("f[1,0](x)"); eval(FromInert(_Inert_FUNCTION(_Inert_NAME("assign"), _Inert_EXPSEQ(ToInert(_ndsol), _Inert_VERBATIM(pointto(_dat[2][3])))))); return FromInert(_Inert_FUNCTION(ToInert(_ndsol), _Inert_EXPSEQ(ToInert(x)))) end if end if; try _res := _solnproc(_xout); _res[3] catch: error end try end proc

 (4)

ftoplot := unapply(f00(x)+0.45e-1*f10(x)/x^(3/2), x)

proc (x) options operator, arrow; f00(x)+0.45e-1*f10(x)/x^(3/2) end proc

 (5)

``

plot(ftoplot(x), x = xini .. xfin)

  

``


Download Test2ODEs.mw

The approach in that file works, however I have a question regarding the efficiency of my method, since I plan to extend the system to many more ODEs besides just 2 and also extend the range to a larger xfin. In this method, since I define the function to plot in terms of f01 and f02, wich are procedures, does this mean that for each x on the grid for the plot(ftoplot,x=xini..xfin) maple actually computes the solutions f00(x) and f01(x) and then forms the ftoplot combination and plots that specific point? If the default sampling of my interval is, say 1000 points, does it mean that the way I wrote it I will have 1000 invocations of the dsolve procedure, for each x in the sample? I am not sure, it seems to me that is the case. This would imply that instead of advancing the solution at each step maple starts over again from xini. How could I just avoid this behavior and instead have access to the values of ftoplot(x) in the range xini to xfin stored from one invocation of dsolve? 

 

The ideal scenario for me would be to have f[0,0](x) and f[0,1](x) stored as an interpolated function between xini and xfin from the solutions of one invocation of dsolve prior to defining ftoplot. Can this be achieved in principle? How? Remember, i have to use method=lsode and range is not accepted.  

 

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