ODE
\[ y''(x)=f(y(x)) \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.905141 (sec), leaf count = 38
\[\text {Solve}\left [\left (c_2+x\right ){}^2=\left (\int _1^{y(x)} \frac {1}{\sqrt {2 \int _1^{K[2]} f(K[1]) \, dK[1]+c_1}} \, dK[2]\right ){}^2,y(x)\right ]\]
Maple ✓
cpu = 0.083 (sec), leaf count = 51
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {2\,\int \!f \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}+{\it \_C1}}}}{d{\it \_b}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt {2\,\int \!f \left ( {\it \_b} \right ) \,{\rm d}{\it \_b}+{\it \_C1}}}}{d{\it \_b}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y''[x] == f[y[x]],y[x],x]
Mathematica raw output
Solve[(x + C[2])^2 == Integrate[1/Sqrt[C[1] + 2*Integrate[f[K[1]], {K[1], 1, K[2]}]], {K[2], 1, y[x]}]^2, y[x]]
Maple raw input
dsolve(diff(diff(y(x),x),x) = f(y(x)), y(x),'implicit')
Maple raw output
Intat(1/(2*Int(f(_b),_b)+_C1)^(1/2),_b = y(x))-x-_C2 = 0, Intat(-1/(2*Int(f(_b),_b)+_C1)^(1/2),_b = y(x))-x-_C2 = 0