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Kitonum

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Ellipses circumscribed around a triangle...

Kitonum 21555
July 26 2020

@one man  There are infinitely many such ellipses, and their family also depends on two parameters. The simple animation below is made for ellipses with one axis parallel to one of the sides of the triangle:


 

> 

restart;
Sol:=solve({x0^2/a^2+y0^2/b^2=1,(2-x0)^2/a^2+(3-y0)^2/b^2=1,(6-x0)^2/a^2+y0^2/b^2=1}, explicit);
sol:=simplify(Sol[1]);

{a = (1/3)*6^(1/2)*((15+(64*b^2+81)^(1/2))/(4*b^2-9))^(1/2)*b, b = b, x0 = 3, y0 = ((3/2)*(15+(64*b^2+81)^(1/2))/(4*b^2-9)-2)*(4*b^2-9)/(15+(64*b^2+81)^(1/2))}, {a = -(1/3)*6^(1/2)*((15+(64*b^2+81)^(1/2))/(4*b^2-9))^(1/2)*b, b = b, x0 = 3, y0 = ((3/2)*(15+(64*b^2+81)^(1/2))/(4*b^2-9)-2)*(4*b^2-9)/(15+(64*b^2+81)^(1/2))}, {a = (1/3)*(-6*(-15+(64*b^2+81)^(1/2))/(4*b^2-9))^(1/2)*b, b = b, x0 = 3, y0 = -(-(3/2)*(-15+(64*b^2+81)^(1/2))/(4*b^2-9)-2)*(4*b^2-9)/(-15+(64*b^2+81)^(1/2))}, {a = -(1/3)*(-6*(-15+(64*b^2+81)^(1/2))/(4*b^2-9))^(1/2)*b, b = b, x0 = 3, y0 = -(-(3/2)*(-15+(64*b^2+81)^(1/2))/(4*b^2-9)-2)*(4*b^2-9)/(-15+(64*b^2+81)^(1/2))}

  

{a = (1/3)*6^(1/2)*((15+(64*b^2+81)^(1/2))/(4*b^2-9))^(1/2)*b, b = b, x0 = 3, y0 = (-16*b^2+3*(64*b^2+81)^(1/2)+81)/(30+2*(64*b^2+81)^(1/2))}

 (1)
> 

plots:-animate(plots:-implicitplot,[eval((x-3)^2/a^2+(y-y0)^2/b^2=1,sol), x=-10..15,y=-10..10, color=red], b=1.55..4, frames=60, background=plots:-display(plottools:-polygon([[0,0],[6,0],[2,3]],color="LightBlue", thickness=2)), scaling=constrained, size=[900,500], axes=none);

 
> 

 


 

Download ells.mw

Link...

Kitonum 21555
July 17 2020
0 0

@gawati2611  See
https://www.mapleprimes.com/questions/226102-Modeling--Of-Inviscid-Fluid-Flow-Around

Re...

Kitonum 21555
July 15 2020
0 0

@Liiiiz 

restart; with(plots): 
A := inequal({am2 < 0.34+3.6*dt}, dt = 0 .. 0.1, am2 = 0 .. 0.7, color = "SkyBlue", numpoints = 8000): 
B := textplot([seq(seq([x, y, "+"], y = 0.03 .. 0.33+3.6*x, 0.03), x = 0.004 .. 0.096, 0.0046)], font = [times, bold, 14]): 
display(A, B);

                

Usual way...

Kitonum 21555
July 11 2020
0 0

@nm  This is not strange and is the usual way in different situations, for example

restart;
solve(sin(x)=1/2, allsolutions);

                         

Nice...

Kitonum 21555
July 10 2020
0 0

@vv  Great idea!  Vote up.

Re...

Kitonum 21555
July 09 2020
0 0

@anthonyfl  I have been working with Maple for about 18 years.

OK...

Kitonum 21555
July 06 2020
0 0

@vv  Thanks for the helpful comment. As you saw, I wrote  "... sometimes ...". Undoubtedly, for examples with  floor , your method is better.

numeric...

Kitonum 21555
June 28 2020
0 0

@mehran rajabi  Maple cannot cope with the symbolic solution of this system, but it can be solved numerically:


 

> 

restart;
eq := diff(F1(zeta), zeta, zeta)-b^2*F1(zeta)+2*exp(-b*zeta)*G1(zeta)+(exp(2*b*zeta)-2*exp(b*zeta)+1)*exp(-2*b*zeta)*(exp(b*zeta)*exp(-2*b*zeta)/(3*b)-(2*(exp(b*zeta)-1))*exp(-2*b*zeta)/(3*b))/(3*b^3)-(exp(b*zeta)-1)^2*(exp(-2*b*zeta))^2/(9*b^4)+(1/3)*(exp(b*zeta)-1)*exp(-2*b*zeta) = 0, diff(G1(zeta), zeta, zeta)-b^2*G1(zeta)-(exp(2*b*zeta)-2*exp(b*zeta)+1)*exp(-2*b*zeta)*exp(-b*zeta)/(3*b^2)-(2*(exp(b*zeta)-1))*exp(-2*b*zeta)*exp(-b*zeta)/(3*b^2)+b^2*exp(-b*zeta) = 0, 2*F1(zeta)+diff(H1(zeta), zeta) = 0;
ics := F1(0) = 0, G1(0) = 0, H1(0) = 0, F1(10) = 0, G1(10) = 0;

diff(diff(F1(zeta), zeta), zeta)-b^2*F1(zeta)+2*exp(-b*zeta)*G1(zeta)+(1/3)*(exp(2*b*zeta)-2*exp(b*zeta)+1)*exp(-2*b*zeta)*((1/3)*exp(b*zeta)*exp(-2*b*zeta)/b-(2/3)*(exp(b*zeta)-1)*exp(-2*b*zeta)/b)/b^3-(1/9)*(exp(b*zeta)-1)^2*(exp(-2*b*zeta))^2/b^4+(1/3)*(exp(b*zeta)-1)*exp(-2*b*zeta) = 0, diff(diff(G1(zeta), zeta), zeta)-b^2*G1(zeta)-(1/3)*(exp(2*b*zeta)-2*exp(b*zeta)+1)*exp(-2*b*zeta)*exp(-b*zeta)/b^2-(2/3)*(exp(b*zeta)-1)*exp(-2*b*zeta)*exp(-b*zeta)/b^2+b^2*exp(-b*zeta) = 0, 2*F1(zeta)+diff(H1(zeta), zeta) = 0

  

F1(0) = 0, G1(0) = 0, H1(0) = 0, F1(10) = 0, G1(10) = 0

 (1)
> 

sol:=dsolve({eval(eq,b=1),ics}, {F1(zeta),G1(zeta),H1(zeta)}, numeric);

proc (x_bvp) local res, data, solnproc, _ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then outpoint := evalf[_EnvDSNumericSaveDigits](x_bvp) else outpoint := evalf(x_bvp) end if; data := Array(1..4, {(1) = proc (outpoint) local X, Y, YP, yout, errproc, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; X := Vector(23, {(1) = .0, (2) = .44928358946598074, (3) = .8992280016236485, (4) = 1.3498361609654799, (5) = 1.8012662223501124, (6) = 2.2538176451082137, (7) = 2.7075058473204168, (8) = 3.1622104441879415, (9) = 3.617523715379814, (10) = 4.073416898400966, (11) = 4.529828534320989, (12) = 4.986426542395182, (13) = 5.443162925516589, (14) = 5.90002876052367, (15) = 6.3569429608286505, (16) = 6.813886254976485, (17) = 7.270857655114633, (18) = 7.727839396874414, (19) = 8.184824918712069, (20) = 8.641783164869185, (21) = 9.097073709220012, (22) = 9.549798354479439, (23) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(23, 5, {(1, 1) = .0, (1, 2) = .1351851752492863, (1, 3) = .0, (1, 4) = .41666665836616046, (1, 5) = .0, (2, 1) = 0.5115494468017586e-1, (2, 2) = 0.8113419738163506e-1, (2, 3) = .11132234411959376, (2, 4) = .12302251075777063, (2, 5) = -0.24914671295543627e-1, (3, 1) = 0.7233554601458965e-1, (3, 2) = 0.17357015828046368e-1, (3, 3) = .13610713779065636, (3, 4) = 0.5134157483476928e-2, (3, 5) = -0.8264444410819323e-1, (4, 1) = 0.7135659319136944e-1, (4, 2) = -0.1696184290536502e-1, (4, 3) = .12674020370180883, (4, 4) = -0.38860099645228664e-1, (4, 5) = -.14854879844833171, (5, 1) = 0.6041349462457967e-1, (5, 2) = -0.28755621998151695e-1, (5, 3) = .10581472844225323, (5, 4) = -0.50409952802245465e-1, (5, 5) = -.20842786708768668, (6, 1) = 0.4709409559748275e-1, (6, 2) = -0.28879081547784926e-1, (6, 3) = 0.8320842921389804e-1, (6, 4) = -0.4811289459656999e-1, (6, 5) = -.25708080263529137, (7, 1) = 0.3492083729507625e-1, (7, 2) = -0.24403957661635056e-1, (7, 3) = 0.6296637788667604e-1, (7, 4) = -0.40707532483008464e-1, (7, 5) = -.2941345144098128, (8, 1) = 0.25070003281839223e-1, (8, 2) = -0.1893111003883e-1, (8, 3) = 0.4638147698328233e-1, (8, 4) = -0.32264795733010464e-1, (8, 5) = -.3212232294743046, (9, 1) = 0.17611571909885645e-1, (9, 2) = -0.13977250930407952e-1, (9, 3) = 0.33493374348802465e-1, (9, 4) = -0.24542586233020944e-1, (9, 5) = -.3404853250515704, (10, 1) = 0.12184701255497917e-1, (10, 2) = -0.10002753559946396e-1, (10, 3) = 0.23817195842104507e-1, (10, 4) = -0.18144557058951072e-1, (10, 5) = -.35393158488637366, (11, 1) = 0.8337091720945352e-2, (11, 2) = -0.70108099469820655e-2, (11, 3) = 0.16729570595585577e-1, (11, 4) = -0.13136581521343518e-1, (11, 5) = -.36319419500666317, (12, 1) = 0.5658426549190852e-2, (12, 2) = -0.4843973820379962e-2, (12, 3) = 0.11636029945898978e-1, (12, 4) = -0.9361364480710756e-2, (12, 5) = -.369509342937797, (13, 1) = 0.3816482825940519e-2, (13, 2) = -0.3312650434613768e-2, (13, 3) = 0.8027270170587932e-2, (13, 4) = -0.6588553177501801e-2, (13, 5) = -.3737837363058767, (14, 1) = 0.25611722459435832e-2, (14, 2) = -0.22483099413305057e-2, (14, 3) = 0.5499046102432881e-2, (14, 4) = -0.4591001213267668e-2, (14, 5) = -.37666052051945037, (15, 1) = 0.17114348068358774e-2, (15, 2) = -0.15174667598063293e-2, (15, 3) = 0.3743981941592335e-2, (15, 4) = -0.3173781454315336e-2, (15, 5) = -.37858736316938824, (16, 1) = 0.11389945207849871e-2, (16, 2) = -0.10202477258979441e-2, (16, 3) = 0.25342610892095207e-2, (16, 4) = -0.21807885842735457e-2, (16, 5) = -.37987258609442215, (17, 1) = 0.7545084866810752e-3, (17, 2) = -0.6846799203792152e-3, (17, 3) = 0.17046420673436833e-2, (17, 4) = -0.1492807199890507e-2, (17, 5) = -.3807262127850646, (18, 1) = 0.4963920372649282e-3, (18, 2) = -0.46017779695480676e-3, (18, 3) = 0.11369868841425213e-2, (18, 4) = -0.10217803121256395e-2, (18, 5) = -.38129005756286605, (19, 1) = 0.3223713762156566e-3, (19, 2) = -0.3118738646949354e-3, (19, 3) = 0.74746077956267e-3, (19, 4) = -0.7044025597402917e-3, (19, 5) = -.3816590721701828, (20, 1) = 0.20336866967754657e-3, (20, 2) = -0.21626740918513138e-3, (20, 3) = 0.4765918146673492e-3, (20, 4) = -0.4965285098485759e-3, (20, 5) = -.38189599488365217, (21, 1) = 0.11931969219112595e-3, (21, 2) = -0.15820939128644842e-3, (21, 3) = 0.2821548893513022e-3, (21, 4) = -0.36905288775930067e-3, (21, 5) = -.3820409112174494, (22, 1) = 0.554102699130244e-4, (22, 2) = -0.12824646354115483e-3, (22, 3) = 0.13198487000106905e-3, (22, 4) = -0.3036077757130274e-3, (22, 5) = -.3821189946686551, (23, 1) = .0, (23, 2) = -0.1216257983196006e-3, (23, 3) = .0, (23, 4) = -0.2913162149422897e-3, (23, 5) = -.3821437169151561}, datatype = float[8], order = C_order); YP := Matrix(23, 5, {(1, 1) = .1351851752492863, (1, 2) = .0, (1, 3) = .41666665836616046, (1, 4) = -1.0, (1, 5) = -.0, (2, 1) = 0.8113419738163506e-1, (2, 2) = -.1645279189060742, (2, 3) = .12302251075777063, (2, 4) = -.40066707748254227, (2, 5) = -.10230988936035172, (3, 1) = 0.17357015828046368e-1, (3, 2) = -.10943416036596883, (3, 3) = 0.5134157483476928e-2, (3, 4) = -.15760241049089635, (3, 5) = -.1446710920291793, (4, 1) = -0.1696184290536502e-1, (4, 2) = -0.46676744936817796e-1, (4, 3) = -0.38860099645228664e-1, (4, 4) = -0.5192526821103144e-1, (4, 5) = -.1427131863827389, (5, 1) = -0.28755621998151695e-1, (5, 2) = -0.9793671716924128e-2, (5, 3) = -0.50409952802245465e-1, (5, 4) = -0.5744900746345538e-2, (5, 5) = -.12082698924915934, (6, 1) = -0.28879081547784926e-1, (6, 2) = 0.6660263410530297e-2, (6, 3) = -0.4811289459656999e-1, (6, 4) = 0.12824170631730872e-1, (6, 5) = -0.941881911949655e-1, (7, 1) = -0.24403957661635056e-1, (7, 2) = 0.11794594639625879e-1, (7, 3) = -0.40707532483008464e-1, (7, 4) = 0.18398806500833406e-1, (7, 5) = -0.698416745901525e-1, (8, 1) = -0.1893111003883e-1, (8, 2) = 0.11760959952295542e-1, (8, 3) = -0.32264795733010464e-1, (8, 4) = 0.1813481396479044e-1, (8, 5) = -0.50140006563678445e-1, (9, 1) = -0.13977250930407952e-1, (9, 2) = 0.9852970906926514e-2, (9, 3) = -0.24542586233020944e-1, (9, 4) = 0.15587535951612403e-1, (9, 5) = -0.3522314381977129e-1, (10, 1) = -0.10002753559946396e-1, (10, 2) = 0.7593608620784867e-2, (10, 3) = -0.18144557058951072e-1, (10, 4) = 0.1246946171648126e-1, (10, 5) = -0.24369402510995834e-1, (11, 1) = -0.70108099469820655e-2, (11, 2) = 0.5580617459638005e-2, (11, 3) = -0.13136581521343518e-1, (11, 4) = 0.9540802895153576e-2, (11, 5) = -0.16674183441890703e-1, (12, 1) = -0.4843973820379962e-2, (12, 2) = 0.39817997532887684e-2, (12, 3) = -0.9361364480710756e-2, (12, 4) = 0.7082571907025149e-2, (12, 5) = -0.11316853098381704e-1, (13, 1) = -0.3312650434613768e-2, (13, 2) = 0.2785777036021733e-2, (13, 3) = -0.6588553177501801e-2, (13, 4) = 0.5143390194185994e-2, (13, 5) = -0.7632965651881038e-2, (14, 1) = -0.22483099413305057e-2, (14, 2) = 0.19223032845477558e-2, (14, 3) = -0.4591001213267668e-2, (14, 4) = 0.36727952288941592e-2, (14, 5) = -0.51223444918871664e-2, (15, 1) = -0.15174667598063293e-2, (15, 2) = 0.13129671199059636e-2, (15, 3) = -0.3173781454315336e-2, (15, 4) = 0.2587539173186767e-2, (15, 5) = -0.34228696136717547e-2, (16, 1) = -0.10202477258979441e-2, (16, 2) = 0.8893352015347192e-3, (16, 3) = -0.21807885842735457e-2, (16, 4) = 0.18019834005339398e-2, (16, 5) = -0.22779890415699743e-2, (17, 1) = -0.6846799203792152e-3, (17, 2) = 0.5975784507371285e-3, (17, 3) = -0.1492807199890507e-2, (17, 4) = 0.12409651380636609e-2, (17, 5) = -0.15090169733621504e-2, (18, 1) = -0.46017779695480676e-3, (18, 2) = 0.3975251515277088e-3, (18, 3) = -0.10217803121256395e-2, (18, 4) = 0.8433904504169269e-3, (18, 5) = -0.9927840745298565e-3, (19, 1) = -0.3118738646949354e-3, (19, 2) = 0.2599871319744332e-3, (19, 3) = -0.7044025597402917e-3, (19, 4) = 0.5615586091124345e-3, (19, 5) = -0.6447427524313132e-3, (20, 1) = -0.21626740918513138e-3, (20, 2) = 0.1639621962441588e-3, (20, 3) = -0.4965285098485759e-3, (20, 4) = 0.3588773029741943e-3, (20, 5) = -0.40673733935509313e-3, (21, 1) = -0.15820939128644842e-3, (21, 2) = 0.94369148619604e-4, (21, 3) = -0.36905288775930067e-3, (21, 4) = 0.20749285308993136e-3, (21, 5) = -0.2386393843822519e-3, (22, 1) = -0.12824646354115483e-3, (22, 2) = 0.3956577750191933e-4, (22, 3) = -0.3036077757130274e-3, (22, 4) = 0.8450778859380169e-4, (22, 5) = -0.1108205398260488e-3, (23, 1) = -0.1216257983196006e-3, (23, 2) = -0.10088873249360564e-4, (23, 3) = -0.2913162149422897e-3, (23, 4) = -0.3026661987284866e-4, (23, 5) = -.0}, datatype = float[8], order = C_order); errproc := proc (x_bvp) local outpoint, X, Y, yout, L, V, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; Digits := 15; outpoint := evalf(x_bvp); X := Vector(23, {(1) = .0, (2) = .44928358946598074, (3) = .8992280016236485, (4) = 1.3498361609654799, (5) = 1.8012662223501124, (6) = 2.2538176451082137, (7) = 2.7075058473204168, (8) = 3.1622104441879415, (9) = 3.617523715379814, (10) = 4.073416898400966, (11) = 4.529828534320989, (12) = 4.986426542395182, (13) = 5.443162925516589, (14) = 5.90002876052367, (15) = 6.3569429608286505, (16) = 6.813886254976485, (17) = 7.270857655114633, (18) = 7.727839396874414, (19) = 8.184824918712069, (20) = 8.641783164869185, (21) = 9.097073709220012, (22) = 9.549798354479439, (23) = 10.0}, datatype = float[8], order = C_order); Y := Matrix(23, 5, {(1, 1) = .0, (1, 2) = -0.5060266839403578e-8, (1, 3) = .0, (1, 4) = -0.5595449761709814e-8, (1, 5) = .0, (2, 1) = -0.4414648603101034e-8, (2, 2) = 0.5237230439637039e-8, (2, 3) = 0.24227823406843136e-8, (2, 4) = -0.3896659498317099e-8, (2, 5) = -0.1830740038095415e-7, (3, 1) = -0.13533808568033192e-8, (3, 2) = 0.20141702480360933e-8, (3, 3) = 0.20179978179724083e-8, (3, 4) = -0.24069379272293536e-8, (3, 5) = -0.14219658207266639e-7, (4, 1) = 0.337331282284249e-9, (4, 2) = -0.68892695799165e-10, (4, 3) = 0.1305133772877301e-8, (4, 4) = -0.1408689414357604e-8, (4, 5) = -0.11353788189169731e-7, (5, 1) = 0.7780950890751212e-9, (5, 2) = -0.6880292639584532e-9, (5, 3) = 0.7758259856104964e-9, (5, 4) = -0.8039695713767156e-9, (5, 5) = -0.10606964433064684e-7, (6, 1) = 0.7062321448788342e-9, (6, 2) = -0.679287934336782e-9, (6, 3) = 0.448310657957495e-9, (6, 4) = -0.45626590029080564e-9, (6, 5) = -0.10778608065549263e-7, (7, 1) = 0.511133376662903e-9, (7, 2) = -0.5038713184705324e-9, (7, 3) = 0.2601727951753781e-9, (7, 4) = -0.2626161291261405e-9, (7, 5) = -0.11168877664051357e-7, (8, 1) = 0.33464351734378353e-9, (8, 2) = -0.3329048397765606e-9, (8, 3) = 0.15569285919298006e-9, (8, 4) = -0.15656106356462702e-9, (8, 5) = -0.11517201941357773e-7, (9, 1) = 0.20863327364243254e-9, (9, 2) = -0.20826607659080967e-9, (9, 3) = 0.9808757002461656e-10, (9, 4) = -0.9846207919438929e-10, (9, 5) = -0.1176552236640131e-7, (10, 1) = 0.12745430400741654e-9, (10, 2) = -0.12737686541825966e-9, (10, 3) = 0.6557519233836531e-10, (10, 4) = -0.6576689298206469e-10, (10, 5) = -0.11925736660809614e-7, (11, 1) = 0.7773030931510619e-10, (11, 2) = -0.7770210481776407e-10, (11, 3) = 0.4623142673833509e-10, (11, 4) = -0.4634130844562985e-10, (11, 5) = -0.12024113981320469e-7, (12, 1) = 0.4793880506691524e-10, (12, 2) = -0.47920081606321304e-10, (12, 3) = 0.3384161881495305e-10, (12, 4) = -0.339098382448478e-10, (12, 5) = -0.12083213151157077e-7, (13, 1) = 0.3010887618409992e-10, (13, 2) = -0.30097400538997424e-10, (13, 3) = 0.25259653273266184e-10, (13, 4) = -0.25306573823900935e-10, (13, 5) = -0.12118669606512674e-7, (14, 1) = 0.19296189453237174e-10, (14, 2) = -0.1929364196429287e-10, (14, 3) = 0.18944755023838608e-10, (14, 4) = -0.1898405470589748e-10, (14, 5) = -0.12140212012100835e-7, (15, 1) = 0.12593525445588906e-10, (15, 2) = -0.12601937695329738e-10, (15, 3) = 0.14135516940578834e-10, (15, 4) = -0.14179273959243492e-10, (15, 5) = -0.12153577902399673e-7, (16, 1) = 0.8329285428099472e-11, (16, 2) = -0.8352481787026178e-11, (16, 3) = 0.10426765405818495e-10, (16, 4) = -0.10488513341902094e-10, (16, 5) = -0.12162075013745392e-7, (17, 1) = 0.5546883414937215e-11, (17, 2) = -0.55922347729653605e-11, (17, 3) = 0.7570774463024126e-11, (17, 4) = -0.7668286628881844e-11, (17, 5) = -0.12167597772930108e-7, (18, 1) = 0.3690750561552212e-11, (18, 2) = -0.3771039349477756e-11, (18, 3) = 0.5388426387445673e-11, (18, 4) = -0.5547066209268148e-11, (18, 5) = -0.12171242390376102e-7, (19, 1) = 0.2427709366136143e-11, (19, 2) = -0.2563746778570831e-11, (19, 3) = 0.373310185312118e-11, (19, 4) = -0.3990516528592753e-11, (19, 5) = -0.12173658534101514e-7, (20, 1) = 0.15486146251726389e-11, (20, 2) = -0.17734116930070722e-11, (20, 3) = 0.24772238266996045e-11, (20, 4) = -0.2890670417236702e-11, (20, 5) = -0.12175239757029905e-7, (21, 1) = 0.9154864023028152e-12, (21, 2) = -0.12804994267799692e-11, (21, 3) = 0.15081801558644428e-11, (21, 4) = -0.2165620235996891e-11, (21, 5) = -0.12176226070623022e-7, (22, 1) = 0.4268778197861308e-12, (22, 2) = -0.10125184044592964e-11, (22, 3) = 0.7188473624212691e-12, (22, 4) = -0.17584063374567492e-11, (22, 5) = -0.12176762317804481e-7, (23, 1) = .0, (23, 2) = -0.9338425049396765e-12, (23, 3) = .0, (23, 4) = -0.16457719352978926e-11, (23, 5) = -0.12176919844781482e-7}, datatype = float[8], order = C_order); if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "right" then return X[23] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(1.830740038095415e-8) elif outpoint = "errorproc" then error "this is already the error procedure" elif outpoint = "rawdata" then return [5, 23, [F1(zeta), diff(F1(zeta), zeta), G1(zeta), diff(G1(zeta), zeta), H1(zeta)], X, Y] else return ('procname')(x_bvp) end if end if; if outpoint < X[1] or X[23] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[23] end if; V := array([1 = 4, 2 = 0]); if Digits <= trunc(evalhf(Digits)) then L := Vector(4, 'datatype' = 'float'[8]); yout := Vector(5, 'datatype' = 'float'[8]); evalhf(`dsolve/numeric/lagrange`(23, 5, X, Y, outpoint, var(yout), var(L), var(V))) else L := Vector(4, 'datatype' = 'sfloat'); yout := Vector(5, 'datatype' = 'sfloat'); `dsolve/numeric/lagrange`(23, 5, X, Y, outpoint, yout, L, V) end if; [zeta = outpoint, seq('[F1(zeta), diff(F1(zeta), zeta), G1(zeta), diff(G1(zeta), zeta), H1(zeta)]'[i] = yout[i], i = 1 .. 5)] end proc; if not type(outpoint, 'numeric') then if outpoint = "start" or outpoint = "left" then return X[1] elif outpoint = "method" then return "bvp" elif outpoint = "right" then return X[23] elif outpoint = "order" then return 8 elif outpoint = "error" then return HFloat(1.830740038095415e-8) elif outpoint = "errorproc" then return eval(errproc) elif outpoint = "rawdata" then return [5, 23, "depnames", X, Y, YP] else error "non-numeric value" end if end if; if outpoint < X[1] or X[23] < outpoint then error "solution is only defined in the range %1..%2", X[1], X[23] end if; if Digits <= trunc(evalhf(Digits)) and (_EnvInFsolve <> true or _EnvDSNumericSaveDigits <= trunc(evalhf(Digits))) then V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0, (3, 1) = .0, (3, 2) = .0, (4, 1) = .0, (4, 2) = .0, (5, 1) = .0, (5, 2) = .0, (6, 1) = .0, (6, 2) = .0, (7, 1) = .0, (7, 2) = .0}, datatype = float[8], order = C_order); yout := Vector(5, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0}, datatype = float[8]); evalhf(`dsolve/numeric/hermite`(23, 5, X, Y, YP, outpoint, var(yout), var(L), var(V))) else if _EnvInFsolve = true then Digits := _EnvDSNumericSaveDigits end if; V := array( 1 .. 6, [( 1 ) = (7), ( 2 ) = (0), ( 3 ) = (false), ( 4 ) = (false), ( 5 ) = (false), ( 6 ) = (false) ] ); L := Matrix(7, 2, {(1, 1) = 0., (1, 2) = 0., (2, 1) = 0., (2, 2) = 0., (3, 1) = 0., (3, 2) = 0., (4, 1) = 0., (4, 2) = 0., (5, 1) = 0., (5, 2) = 0., (6, 1) = 0., (6, 2) = 0., (7, 1) = 0., (7, 2) = 0.}, order = C_order); yout := Vector(5, {(1) = 0., (2) = 0., (3) = 0., (4) = 0., (5) = 0.}); `dsolve/numeric/hermite`(23, 5, X, Y, YP, outpoint, yout, L, V) end if; [outpoint, seq(yout[i], i = 1 .. 5)] end proc, (2) = Array(0..0, {}), (3) = [zeta, F1(zeta), diff(F1(zeta), zeta), G1(zeta), diff(G1(zeta), zeta), H1(zeta)], (4) = 0}); solnproc := data[1]; if not type(outpoint, 'numeric') then if outpoint = "solnprocedure" then return eval(solnproc) elif member(outpoint, ["start", "left", "right", "errorproc", "rawdata", "order", "error"]) then return solnproc(x_bvp) elif outpoint = "sysvars" then return data[3] elif procname <> unknown then return ('procname')(x_bvp) else _ndsol := pointto(data[2][0]); return ('_ndsol')(x_bvp) end if end if; try res := solnproc(outpoint); [zeta = res[1], seq('[F1(zeta), diff(F1(zeta), zeta), G1(zeta), diff(G1(zeta), zeta), H1(zeta)]'[i] = res[i+1], i = 1 .. 5)] catch: error end try end proc

 (2)
> 

plots:-odeplot(sol,[[zeta,F1(zeta)],[zeta,G1(zeta)],[zeta,H1(zeta)]], zeta=0..10, color=[red,blue,green], size=[500,500]);

 
> 

 


 

Download numeric.mw

assuming...

Kitonum 21555
June 28 2020
0 0

@mehran rajabi  Add the line

sol:=simplify(sol) assuming b>0;

explicit...

Kitonum 21555
June 23 2020

Try the option  explicit  in the last line:

T:=solve({f1,f2,f3}, [x,y1,y2], explicit);

 

?...

Kitonum 21555
June 21 2020
0 0

@spinoza  I did not understand the meaning of your question.

No...

Kitonum 21555
June 17 2020
0 0

@Carl Love  Your plot  does not prove anything, because it is built on a finite set of points, and the break point is somewhere between them. Here is another plot  on a very small interval, inside which there is a break point (the zero of the denominator):

Digits:=20:
fsolve(denom(x(t)));
plot(x(t), t=-0.00077376199177721..-0.00077376199177719, -5000..5000);

      

 

Adjustment...

Kitonum 21555
June 16 2020
0 0

@666 jvbasha 


 

> 

restart;
ContoursWithLabels := proc (Expr, Range1::(range(realcons)), Range2::(range(realcons)), Number::posint := 8, S::(set(realcons)) := {}, GraphicOptions::list := [color = black, axes = box], Coloring::`=` := NULL)
local r1, r2, L, f, L1, h, S1, P, P1, r, M, C, T, p, p1, m, n, A, B, E;
uses plots, plottools;
f := unapply(Expr, x, y);
if S = {} then r1 := rand(convert(Range1, float)); r2 := rand(convert(Range2, float));
L := [seq([r1(), r2()], i = 1 .. 205)];
L1 := convert(sort(select(a->type(a, realcons), [seq(f(op(t)), t = L)]), (a, b) ->is(abs(a) < abs(b))), set);
h := (L1[-6]-L1[1])/Number;
S1 := [seq(L1[1]+(1/2)*h+h*(n-1), n = 1 .. Number)] else
S1 := convert(S, list)  fi;
print(Contours = evalf[2](S1));
r := k->rand(20 .. k-20); M := []; T := [];
for C in S1 do
P := implicitplot(Expr = C, x = Range1, y = Range2, op(GraphicOptions), gridrefine = 3);
P1 := [getdata(P)];
for p in P1 do
p1 := convert(p[3], listlist); n := nops(p1);
if n < 500 then m := `if`(40 < n, (r(n))(), round((1/2)*n)); M := `if`(40 < n, [op(M), p1[1 .. m-11], p1[m+11 .. n]], [op(M), p1]); T := [op(T), [op(p1[m]), evalf[2](C)]] else
if 500 <= n then h := floor((1/2)*n); m := (r(h))(); M := [op(M), p1[1 .. m-11], p1[m+11 .. m+h-11], p1[m+h+11 .. n]]; T := [op(T), [op(p1[m]), evalf[2](C)], [op(p1[m+h]), evalf[2](C)]]
fi; fi; od; od;
A := plot(M, op(GraphicOptions));
B := plots:-textplot(T);
if Coloring = NULL then E := NULL else E := ([plots:-densityplot])(Expr, x = Range1, y = Range2, op(GraphicOptions), op(rhs(Coloring)))  fi;
display(E, A, B);
end proc:
> 

# Your new example

A3 := .25*y*(-6*x^2+6*x-2.477250468*x*(x-1)^2-2.477250468*x^2*(x-1)-1.476663599*x*(x-1)^3-2.214995399*x^2*(x-1)^2+.3837076420*x*(x-1)^4+.7674152840*x^2*(x-1)^3+1.049305257*x*(x-1)^5+2.623263142*x^2*(x-1)^4+1.470504325*x*(x-1)^6+4.411512974*x^2*(x-1)^5+2.062933702*x*(x-1)^7+7.220267957*x^2*(x-1)^6+1.610136961*x*(x-1)^8+6.440547843*x^2*(x-1)^7+.6577852166*x*(x-1)^9+2.960033475*x^2*(x-1)^8);




ContoursWithLabels(A3, 0 .. 1, 0 .. 10, {seq(0.1..5,0.2)}, [color = black, thickness = 2, axes = box, size=[450,450], labels = ["eta", "r"],labeldirections = [horizontal, vertical], labelfont = ['TIMES', 'BOLDOBLIQUE', 16]], Coloring = [colorstyle = HUE,style = surface]);

.25*y*(-6*x^2+6*x-2.477250468*x*(x-1)^2-2.477250468*x^2*(x-1)-1.476663599*x*(x-1)^3-2.214995399*x^2*(x-1)^2+.3837076420*x*(x-1)^4+.7674152840*x^2*(x-1)^3+1.049305257*x*(x-1)^5+2.623263142*x^2*(x-1)^4+1.470504325*x*(x-1)^6+4.411512974*x^2*(x-1)^5+2.062933702*x*(x-1)^7+7.220267957*x^2*(x-1)^6+1.610136961*x*(x-1)^8+6.440547843*x^2*(x-1)^7+.6577852166*x*(x-1)^9+2.960033475*x^2*(x-1)^8)

  

Contours = [.1, .3, .5, .7, .9, 1.1, 1.3, 1.5, 1.7, 1.9, 2.1, 2.3, 2.5, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1, 4.3, 4.5, 4.7, 4.9]

  

 
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Download ContoursWithLabels11.mw

Example...

Kitonum 21555
May 27 2020
0 0

@radaar  For example, when you define a procedure using  arrow-notation:

x:=10: t:=20:
f:=x->x^2:
g:=t->t^2:
f(3); g(3);

f  and  g  are  the same function.

Automated...

Kitonum 21555
May 27 2020
0 0

The solution to similar and much more complex examples is automated in this post
https://www.mapleprimes.com/posts/145922-Perimeter-Area-And-Visualization-Of-A-Plane-Figure-

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