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 = 0.197821 (sec), leaf count = 40
\[\text {Solve}\left [(x+c_2){}^2=\int _1^{y(x)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}f(K[1])dK[1]}}dK[2]{}^2,y(x)\right ]\]
Maple ✓
cpu = 0.444 (sec), leaf count = 51
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )+\textit {\_C1}}}d \textit {\_b} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )+\textit {\_C1}}}d \textit {\_b} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y''[x] == f[y[x]],y[x],x]
Mathematica raw output
Solve[(x + C[2])^2 == Inactive[Integrate][1/Sqrt[C[1] + 2*Inactive[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))
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]