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How to sovle the second order ODE numerically？

Daniel Niu

Posted 3 years ago

I solved as follows: eqn = {y''[x] == x^(-1/2)*y[x]^(3/2), y[0] == 1, y[∞] ==0}; f1 = NDSolveValue[eqn, y, {x, 0.1, 5}] ListPlot[Table[{x, f1[x]}, {x, 0, 5, 0.1}], PlotLegends -> {"numerically"}] However, I get nothing. Your help would be highly appreciated!

POSTED BY: Daniel Niu

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Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 3 years ago

Looks like this ODE is very hard to numerically integrate. f = With[{e = 10^-30, INF = 45}, NDSolveValue[{y''[x]Sqrt[x] == (y[x]^(3/2) // Re), y[e] == 1, y[INF] == 0}, y, {x, 0, INF}, Method -> {"BoundaryValues" -> {"Shooting", "StartingInitialConditions" -> {y[e] == 1, y'[e] == -(186757417/117599387)}}}, MaxSteps -> 10^6, WorkingPrecision -> 20]](Only Real part*) Plot[{f[x], f'[x]}, {x, 10^-30, 45}, PlotRange -> All]

Looks like this ODE is very hard to numerically integrate.

f = With[{e = 10^-30, INF = 45}, 
  NDSolveValue[{y''[x]*Sqrt[x] == (y[x]^(3/2) // Re), y[e] == 1, 
    y[INF] == 0}, y, {x, 0, INF}, 
   Method -> {"BoundaryValues" -> {"Shooting", 
       "StartingInitialConditions" -> {y[e] == 1, 
         y'[e] == -(186757417/117599387)}}}, MaxSteps -> 10^6, 
   WorkingPrecision -> 20]](*Only Real part*)

Plot[{f[x], f'[x]}, {x, 10^-30, 45}, PlotRange -> All]

POSTED BY: Mariusz Iwaniuk