4.39.40 \((x-y(x)) y''(x)=f\left (y'(x)\right )\)

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 = 0.845814 (sec), leaf count = 73

\[\text {Solve}\left [\left \{x=\int \frac {\exp \left (-\int _1^{\text {K$\$$310978}} \frac {K[3]-1}{f(K[3])} \, dK[3]-c_1\right )}{f(\text {K$\$$310978})} \, d\text {K$\$$310978}+c_2,x=\exp \left (-\int _1^{\text {K$\$$310978}} \frac {K[3]-1}{f(K[3])} \, dK[3]-c_1\right )+y(x)\right \},\{y(x),\text {K$\$$310978}\}\right ]\]

Maple ✓
cpu = 0.152 (sec), leaf count = 42

\[ \left \{ x-\int ^{y \left ( x \right ) -x}\! \left ( -1+{\it RootOf} \left ( \int ^{{\it \_Z}}\!{\frac {{\it \_a}-1}{f \left ( {\it \_a} \right ) }}{d{\it \_a}}+\ln \left ( -{\it \_g} \right ) +{\it \_C1} \right ) \right ) ^{-1}{d{\it \_g}}-{\it \_C2}=0 \right \} \] 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] - Integrate[(-1 + K[3])/f[K[3]], {K[3], 1,
 K$310978}])/f[K$310978], K$310978], x == E^(-C[1] - Integrate[(-1 + K[3])/f[K[3
]], {K[3], 1, K$310978}]) + y[x]}, {y[x], K$310978}]

Maple raw input

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

Maple raw output

x-Intat(1/(-1+RootOf(Intat(1/f(_a)*(_a-1),_a = _Z)+ln(-_g)+_C1)),_g = y(x)-x)-_C
2 = 0