ODE
\[ (x-y(x)) y''(x)=f\left (y'(x)\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.465522 (sec), leaf count = 80
\[\text {Solve}\left [\left \{x=\int \frac {\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{f(K[3])}dK[3]-c_1\right )}{f(K[4])} \, dK[4]+c_2,x=y(x)+\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{f(K[3])}dK[3]-c_1\right )\right \},\{y(x),K[4]\}\right ]\]
Maple ✓
cpu = 0.973 (sec), leaf count = 39
\[\left [y \left (x \right ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {1}{-1+\RootOf \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a} -1}{f \left (\textit {\_a} \right )}d \textit {\_a} +\ln \left (-\textit {\_g} \right )+\textit {\_C1} \right )}d \textit {\_g} +\textit {\_C2} \right )\right ]\] Mathematica raw input
DSolve[(x - y[x])*y''[x] == f[y'[x]],y[x],x]
Mathematica raw output
Solve[{x == C[2] + Integrate[E^(-C[1] - Inactive[Integrate][(-1 + K[3])/f[K[3]], {K[3], 1, K[4]}])/f[K[4]], K[4]], x == E^(-C[1] - Inactive[Integrate][(-1 + K[3])/f[K[3]], {K[3], 1, K[4]}]) + y[x]}, {y[x], K[4]}]
Maple raw input
dsolve((x-y(x))*diff(diff(y(x),x),x) = f(diff(y(x),x)), y(x))
Maple raw output
[y(x) = x+RootOf(-x+Intat(1/(-1+RootOf(Intat(1/f(_a)*(_a-1),_a = _Z)+ln(-_g)+_C1)),_g = _Z)+_C2)]