4
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I am trying to find an exact solution to this differential equation,

ode = f'''[x] + f[x]*f''[x] - f'[x]^2 == 0

For which, I already know the exact solution exist in the form,

Exactsol = (1 - Exp[-x])

Simplify[ode /. {{f[x] -> %}} /. {{f'[x] -> D[%, x]}} /. {{f''[x] -> 
       D[%, x, x]}} /. {{f'''[x] -> D[%, x, x, x]}}] // Flatten

But DSolve is unable to find any closed form solution to this ode.

Any thoughts?

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  • 2
    $\begingroup$ Exactsol = (-r - Exp[r x]) also produces True. $\endgroup$ – LouisB Apr 15 '17 at 9:46
  • 3
    $\begingroup$ A still more general solution is Exactsol = c1 + c2 Exp[-c1 x]. $\endgroup$ – bbgodfrey Apr 15 '17 at 21:03

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