How to sovle the second order ODE numerically？

Questions
, ,
1

145.png
145.png849×334 21.9 KB

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!

2

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]
3

Thank you, Mariusz ! The results are extremely helpful.

4

Dear Mariusz,
Would you mind giving me more explanation how to get y’[e] == -(186757417/117599387)
Thank you so much!
Best regards!

5

With trial and error.