ODE
\[ 2 a^2 y(x)^2+y(x) y''(x)+y'(x)^2=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.125994 (sec), leaf count = 23
\[\left \{\left \{y(x)\to c_2 \sqrt {\cos \left (a \left (2 x-c_1\right )\right )}\right \}\right \}\]
Maple ✓
cpu = 0.064 (sec), leaf count = 26
\[ \left \{ -a \left ( y \left ( x \right ) \right ) ^{2}-{\it \_C1}\,\sin \left ( 2\,ax \right ) +{\it \_C2}\,\cos \left ( 2\,ax \right ) =0 \right \} \] Mathematica raw input
DSolve[2*a^2*y[x]^2 + y'[x]^2 + y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2]*Sqrt[Cos[a*(2*x - C[1])]]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2+2*a^2*y(x)^2 = 0, y(x),'implicit')
Maple raw output
-a*y(x)^2-_C1*sin(2*a*x)+_C2*cos(2*a*x) = 0