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 = 0.406624 (sec), leaf count = 23
\[\left \{\left \{y(x)\to c_2 \sqrt {\cos (a (2 x-c_1))}\right \}\right \}\]
Maple ✓
cpu = 0.614 (sec), leaf count = 58
\[\left [y \left (x \right ) = \frac {\sqrt {-a \left (\textit {\_C1} \sin \left (2 a x \right )-\textit {\_C2} \cos \left (2 a x \right )\right )}}{a}, y \left (x \right ) = -\frac {\sqrt {-a \left (\textit {\_C1} \sin \left (2 a x \right )-\textit {\_C2} \cos \left (2 a x \right )\right )}}{a}\right ]\] 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))
Maple raw output
[y(x) = 1/a*(-a*(_C1*sin(2*a*x)-_C2*cos(2*a*x)))^(1/2), y(x) = -1/a*(-a*(_C1*sin(2*a*x)-_C2*cos(2*a*x)))^(1/2)]