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 = 0.110461 (sec), leaf count = 34
\[\left \{\left \{y(x)\to \frac {c_2 e^{\sqrt {2} x}}{e^{2 \sqrt {2} \left (c_1+x\right )}+1}\right \}\right \}\]
Maple ✓
cpu = 0.039 (sec), leaf count = 30
\[ \left \{ -{{\rm e}^{2\,\sqrt {2}x}}{\it \_C1}+2\,{\frac {\sqrt {2}{{\rm e}^{\sqrt {2}x}}}{y \left ( x \right ) }}+{\it \_C2}=0 \right \} \] 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),'implicit')
Maple raw output
-exp(2*2^(1/2)*x)*_C1+2*2^(1/2)*exp(2^(1/2)*x)/y(x)+_C2 = 0