ODE
\[ y(x) y''(x)=y'(x)^2+y(x)^2 \log (y(x)) \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.55138 (sec), leaf count = 73
\[\left \{\left \{y(x)\to \exp \left (-\frac {1}{2} \sqrt {c_1} e^{-x-c_2} \left (-1+e^{2 (x+c_2)}\right )\right )\right \},\left \{y(x)\to \exp \left (\frac {1}{2} \sqrt {c_1} e^{-x-c_2} \left (-1+e^{2 (x+c_2)}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 1.178 (sec), leaf count = 21
\[\left [y \left (x \right ) = {\mathrm e}^{\frac {{\mathrm e}^{-2 x} \textit {\_C1} \,{\mathrm e}^{x}}{2}} {\mathrm e}^{-\frac {\textit {\_C2} \,{\mathrm e}^{x}}{2}}\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == Log[y[x]]*y[x]^2 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> E^(-1/2*(E^(-x - C[2])*(-1 + E^(2*(x + C[2])))*Sqrt[C[1]]))}, {y[x] -> E^((E^(-x - C[2])*(-1 + E^(2*(x + C[2])))*Sqrt[C[1]])/2)}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+y(x)^2*ln(y(x)), y(x))
Maple raw output
[y(x) = exp(1/2*exp(-2*x)*_C1*exp(x))*exp(-1/2*_C2*exp(x))]