The equation that I need an analytical solution of, is
f'''= \frac{1}{3}f'^2+\frac{2}{3}ff''
The boundary conditions are:
at x=0; f=0, f'=1
at x=\infty; f'=0
If I use Dsolve, it doesn’t return a solution
sol = Dsolve[y'''[x] == (1/3)*(y'[x])^2 - (2/3)*(y[x])*(y''[x]), y[0] == 0, y'[0] = 1, y'[inf] == 0, y[x], x]
Likely, that means Mathematica doesn’t know a closed-form solution, if it exists.
DSolve[{f'''[x] == 1/3*f'[x]^2 - 2/3*f[x]*f''[x], f[0] == 0, f'[0] == 1, f'[Infinity] == 0}, f[x], x]
(* Returns unevaluated *)
Numeric method:
sol = NDSolve[{f'''[x] == 1/3*f'[x]^2 - 2/3*f[x]*f''[x], f[0] == 0, f'[0] == 1, f'[20] == 0}, f, {x, 0, 20}]
Plot[{f[x] /. sol, Evaluate@D[f[x] /. sol, x]}, {x, 0, 20}, PlotLegends -> {"f[x]", "f'[x]"}]
For analytic solution, you may try from here and use Mathematica to find a solution from lower degree.
Thanks, the numerical solution is what I expected it to be.
It can be reduced to second-order ODE with substitution:
f'''(x)=u''(f) u(f)^2+u'(f)^2 u(f)
f''(x)=u(f) u'(f)
f'(x)=u(f) where f(x)=f
Putting to equation:
u''[f]*u[f]^2 + u'[f]^2*u[f] == 1/3*u[f]^2 - 2/3*f*u[f]*u'[f]
u''[f]*u[f] + u'[f]^2 == 1/3*u[f] - 2/3*f*u'[f]
eq2 = u''[f]*u[f] + u'[f]^2 + 2/3*f*u'[f] - 1/3*u[f] == 0
DSolve[eq2, u[f], f]
(* Returns unevaluated *)
With help of Maple it can be reduced to Abel second kind equation,but DSolve returns unevaluated.
DSolve[y'[x]*y[x] - x^2/3 + 2/3*y[x] == 0, y[x], x]
( * Input ?*)
I can only say Mathematica 11.3 and Maple 2018 doesn’t know a closed-form solution.