ODE
\[ y(x) y''(x)=2 \left (y'(x)^2-y(x)^2\right ) \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.342345 (sec), leaf count = 34
\[\left \{\left \{y(x)\to \frac {c_2 e^{\sqrt {2} x}}{1+e^{2 \sqrt {2} (x+c_1)}}\right \}\right \}\]
Maple ✓
cpu = 0.458 (sec), leaf count = 30
\[\left [y \left (x \right ) = \frac {2 \,{\mathrm e}^{\sqrt {2}\, x} \sqrt {2}}{{\mathrm e}^{2 \sqrt {2}\, x} \textit {\_C1} -\textit {\_C2}}\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == 2*(-y[x]^2 + y'[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> (E^(Sqrt[2]*x)*C[2])/(1 + E^(2*Sqrt[2]*(x + C[1])))}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = 2*diff(y(x),x)^2-2*y(x)^2, y(x))
Maple raw output
[y(x) = 2*exp(2^(1/2)*x)*2^(1/2)/(exp(2*2^(1/2)*x)*_C1-_C2)]