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This is my 3rd Order IVP. D[y[x], {x, 3}] == (m/2)*((D[y[x], x])^2 - 1) - ((m + 1)/2)* y*(D[y[x], {x, 2}])^2. Initial conditions given are as follows y[0] = 0; y'[0] = 0; y''[0] = 1; (* This is the First assumed value of y''[0] for initializing shooting method using RK 4rth order method *)

For this problem y'[infinity] is equal to 1, I want to loop the shooting method using Runge Kutta 4 th order method in such a way that after calculations, it will check the y'[] value at sufficiently large value of x and from experimental results, for x ~ 7-8, f'[] is 1.

To Use RK method I redefined the equation like

D[l[x], {x, 2}] ==  z; D[y[x], x] ==  l; D[z[x], x] == (m/2)*(l^2 - 1) - ((m + 1)/2)*z*y;

Now I could write one loop for RK method but this is proving out a bit tough for me because I have to generate a list for different values of f[0] { 0,1} and m is also a variable with range (0-1) step size 0.01 & then I have to deal with 3 equations simultaneously. I am solving this over C programming but its not that efficient. I dont know how to go ahead. If anyone could throw some light.....

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Where's the definition of m? Then your equation involves simple mistake, y*(D[y[x], {x, 2}])^2 should be y[x] (D[y[x], {x, 2}])^2. Finally, I tried to set m = 1, and this equation is easily solved by NDSolve. –  xzczd Mar 3 at 2:17
    
Have there been some recent homework assignments on the shooting method? Seems to have been a few related questions recently. Is this a homework problem? –  Mike Honeychurch Mar 3 at 4:17
    
@xzczd- How could you solve this ODE with NDsolve. What you assumed the value of f''[0]. Since we only know f[0] and f'[0]. We do know f'[infinity]= 1 but we cant use this condition in NDSolve, that's why we have to use shooting method I guess. m is a list you can assume - {0.1,0.2,0.3...........1.0}. f[0] is same list. I wish to make a Module function which takes these euqation, perform RK method over them and solved and plot the results for different m and f[0]. –  user11948 Mar 3 at 6:02
    
@MikeHoneychurch- I dont know about any homework but this a problem related to Boundary layer flow with Wedge effect and Blowing and suction together. –  user11948 Mar 3 at 6:03
    
@MikeHoneychurch this also is applied as an approximation method in quantum chemistry. It converges slowly. –  olliepower Mar 3 at 6:30

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