ODE
\[ f\left (y(x),y''(x)\right )=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 8.75089 (sec), leaf count = 42
\[\text {Solve}\left [\left (c_2+x\right ){}^2=\left (\int _1^{y(x)} \frac {1}{\sqrt {2 \int _1^{K[2]} \text {InverseFunction}[f,2,2][K[1],0] \, dK[1]+c_1}} \, dK[2]\right ){}^2,y(x)\right ]\]
Maple ✓
cpu = 0.086 (sec), leaf count = 33
\[ \left \{ \int ^{y \left ( x \right ) }\! \left ( {\it RootOf} \left ( {{\it \_Z}}^{2}-{\it \_C1}-2\,\int \!{\it RootOf} \left ( f \left ( {\it \_b},{\it \_Z} \right ) \right ) \,{\rm d}{\it \_b} \right ) \right ) ^{-1}{d{\it \_b}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[f[y[x], y''[x]] == 0,y[x],x]
Mathematica raw output
Solve[(x + C[2])^2 == Integrate[1/Sqrt[C[1] + 2*Integrate[InverseFunction[f, 2, 2][K[1], 0], {K[1], 1, K[2]}]], {K[2], 1, y[x]}]^2, y[x]]
Maple raw input
dsolve(f(y(x),diff(diff(y(x),x),x)) = 0, y(x),'implicit')
Maple raw output
Intat(1/RootOf(_Z^2-_C1-2*Int(RootOf(f(_b,_Z)),_b)),_b = y(x))-x-_C2 = 0