ODE
\[ y''(x)=a \sqrt {b y(x)^2+y'(x)^2} \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✗
cpu = 0.723433 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == a*Sqrt[b*y[x]^2 + Derivative[1][y][x]^2], y[x], x]
Maple ✓
cpu = 0.314 (sec), leaf count = 41
\[ \left \{ \ln \left ( y \left ( x \right ) \right ) -\int \!{\it RootOf} \left ( x-\int ^{{\it \_Z}}\! \left ( a\sqrt {{{\it \_f}}^{2}+b}-{{\it \_f}}^{2} \right ) ^{-1}{d{\it \_f}}+{\it \_C1} \right ) \,{\rm d}x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y''[x] == a*Sqrt[b*y[x]^2 + y'[x]^2],y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == a*Sqrt[b*y[x]^2 + Derivative[1][y][x]^2], y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(b*y(x)^2+diff(y(x),x)^2)^(1/2), y(x),'implicit')
Maple raw output
ln(y(x))-Int(RootOf(x-Intat(1/(a*(_f^2+b)^(1/2)-_f^2),_f = _Z)+_C1),x)-_C2 = 0