4.37.39 \(y''(x)=f\left (y'(x)\right )\)

ODE
\[ y''(x)=f\left (y'(x)\right ) \] ODE Classification

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.103086 (sec), leaf count = 33

\[\left \{\left \{y(x)\to \int _1^x \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{f(K[1])} \, dK[1]\& \right ]\left [K[2]+c_1\right ] \, dK[2]+c_2\right \}\right \}\]

Maple ✓
cpu = 0.041 (sec), leaf count = 22

\[ \left \{ y \left ( x \right ) =\int \!{\it RootOf} \left ( x-\int ^{{\it \_Z}}\! \left ( f \left ( {\it \_f} \right ) \right ) ^{-1}{d{\it \_f}}+{\it \_C1} \right ) \,{\rm d}x+{\it \_C2} \right \} \] Mathematica raw input

DSolve[y''[x] == f[y'[x]],y[x],x]

Mathematica raw output

{{y[x] -> C[2] + Integrate[InverseFunction[Integrate[f[K[1]]^(-1), {K[1], 1, #1}
] & ][C[1] + K[2]], {K[2], 1, x}]}}

Maple raw input

dsolve(diff(diff(y(x),x),x) = f(diff(y(x),x)), y(x),'implicit')

Maple raw output

y(x) = Int(RootOf(x-Intat(1/f(_f),_f = _Z)+_C1),x)+_C2