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Mathematics Calculus Equation Solving Wolfram Language

Get an analytical solution of a non-linear 3rd order ode?

Pragyan Sarma

Posted 8 years ago

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]

POSTED BY: Pragyan Sarma

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Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 8 years ago

Likely, that means Mathematica doesn't know a closed-form solution, if it exists. DSolve[{f'''[x] == 1/3f'[x]^2 - 2/3f[x]f''[x], f[0] == 0, f'[0] == 1, f'[Infinity] == 0}, f[x], x] ( Returns unevaluated ) Numeric method: sol = NDSolve[{f'''[x] == 1/3f'[x]^2 - 2/3f[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]"}]

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]"}]

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