ODE
\[ y(x) y''(x)+y'(x)^2=a^2 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0622942 (sec), leaf count = 64
\[\left \{\left \{y(x)\to -\frac {\sqrt {a^4 \left (c_2+x\right ){}^2-e^{2 c_1}}}{a}\right \},\left \{y(x)\to \frac {\sqrt {a^4 \left (c_2+x\right ){}^2-e^{2 c_1}}}{a}\right \}\right \}\]
Maple ✓
cpu = 0.048 (sec), leaf count = 23
\[ \left \{ -{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2}}+{\frac {{a}^{2}{x}^{2}}{2}}-{\it \_C1}\,x+{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y'[x]^2 + y[x]*y''[x] == a^2,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[-E^(2*C[1]) + a^4*(x + C[2])^2]/a)}, {y[x] -> Sqrt[-E^(2*C[1]) + a^4*(x + C[2])^2]/a}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2 = a^2, y(x),'implicit')
Maple raw output
-1/2*y(x)^2+1/2*a^2*x^2-_C1*x+_C2 = 0