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 = 124.833 (sec), leaf count = 74
\[\text {Solve}\left [\int _1^x -\frac {c_1 e^{K[2]}+K[2]+1}{c_1 e^{K[2]} K[2]+2 K[2]^2+K[2]} \, dK[2]+c_2=\int _1^{y(x)} -\frac {x}{c_1 e^{x K[1]}+2 x K[1]+1} \, dK[1],y(x)\right ]\]
Maple ✓
cpu = 0.273 (sec), leaf count = 37
\[ \left \{ \ln \left ( x \right ) -{\it \_C2}+\int ^{xy \left ( x \right ) }\!- \left ( {{\rm e}^{-{\it \_f}}}{{\rm e}^{{\it \_f}}} \left ( {\it \_f}+1 \right ) -{{\rm e}^{{\it \_f}}}{\it \_C1}+{\it \_f} \right ) ^{-1}{d{\it \_f}}=0 \right \} \] 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] + 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