ODE
\[ x y''(x)=x^2 y'(x)^2-2 y'(x)-y(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.751109 (sec), leaf count = 136
\[\text {Solve}\left [\int _1^x\left (\int _1^{y(x)}\frac {e^{K[1] K[2]} c_1 (K[1] K[2]-1)-1}{\left (e^{K[1] K[2]} c_1+2 K[1] K[2]+1\right ){}^2}dK[1]-\frac {e^{K[2] y(x)} c_1+K[2] y(x)+1}{2 y(x) K[2]^2+e^{K[2] y(x)} c_1 K[2]+K[2]}\right )dK[2]+c_2=\int _1^{y(x)}-\frac {x}{e^{x K[1]} c_1+2 x K[1]+1}dK[1],y(x)\right ]\]
Maple ✓
cpu = 0.406 (sec), leaf count = 32
\[\left [y \left (x \right ) = \frac {\RootOf \left (-\ln \left (x \right )+\textit {\_C2} +\int _{}^{\textit {\_Z}}-\frac {1}{{\mathrm e}^{\textit {\_f}} \textit {\_C1} -2 \textit {\_f} -1}d \textit {\_f} \right )}{x}\right ]\] 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] + Inactive[Integrate][-((1 + E^(K[2]*y[x])*C[1] + K[2]*y[x])/(K[2] + E^(K[2]*y[x])*C[1]*K[2] + 2*K[2]^2*y[x])) + Inactive[Integrate][(-1 + E^(K[1]*K[2])*C[1]*(-1 + K[1]*K[2]))/(1 + E^(K[1]*K[2])*C[1] + 2*K[1]*K[2])^2, {K[1], 1, y[x]}], {K[2], 1, x}] == Inactive[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))
Maple raw output
[y(x) = RootOf(-ln(x)+_C2+Intat(-1/(exp(_f)*_C1-2*_f-1),_f = _Z))/x]