4.38.7 $x{y}^{″}(x)=x^2{y}^{\prime }(x{)}^2-2y^{\prime }(x)-y(x{)}^2$

ODE
$$x{y}^{″}(x)=x^2{y}^{\prime }(x{)}^2-2y^{\prime }(x)-y(x{)}^2$$ ODE Classification

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 124.833 (sec), leaf count = 74

$$\text{Solve}[{\int }_1^x-\frac{c_1{e}^{K[2]}+K[2]+1}{c_1{e}^{K[2]}K[2]+2K[2{]}^2+K[2]}dK[2]+c_2={\int }_1^{y(x)}-\frac{x}{c_1{e}^{xK[1]}+2xK[1]+1}dK[1],y(x)]$$

Maple ✓
cpu = 0.273 (sec), leaf count = 37

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{2}+{\int }^{xy\left(x\right)}-{({\mathrm{e}}^{-\mathit{\_}\mathit{f}}{\mathrm{e}}^{\mathit{\_}\mathit{f}}(\mathit{\_}\mathit{f}+1)-{\mathrm{e}}^{\mathit{\_}\mathit{f}}\mathit{\_}\mathit{C}\mathit{1}+\mathit{\_}\mathit{f})}^{-1}d\mathit{\_}\mathit{f}=0\}$$ Mathematica raw input

DSolve[x*y''[x] == -y[x]^2 - 2*y'[x] + x^2*y'[x]^2,y[x],x]

Mathematica raw output

Solve[C[2] + Integrate[-((1 + E^K[2]*C[1] + K[2])/(K[2] + E^K[2]*C[1]*K[2] + 2*K
[2]^2)), {K[2], 1, x}] == Integrate[-(x/(1 + E^(x*K[1])*C[1] + 2*x*K[1])), {K[1]
, 1, y[x]}], y[x]]

Maple raw input

dsolve(x*diff(diff(y(x),x),x) = x^2*diff(y(x),x)^2-2*diff(y(x),x)-y(x)^2, y(x),'implicit')

Maple raw output

ln(x)-_C2+Intat(-1/(exp(-_f)*exp(_f)*(_f+1)-exp(_f)*_C1+_f),_f = x*y(x)) = 0