ODE
\[ x^4 y''(x)=x^2 y'(x) \left (y'(x)+x\right )-4 y(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 297.41 (sec), leaf count = 89
\[\text {Solve}\left [\int _1^x \frac {K[2]^2 \left (2-c_1 e^{\frac {1}{K[2]^2}}\right )+2}{K[2] \left (K[2]^2 \left (c_1 e^{\frac {1}{K[2]^2}}-2\right )-4\right )} \, dK[2]+c_2=\int _1^{y(x)} \frac {1}{x^2 \left (2-c_1 e^{\frac {K[1]}{x^2}}\right )+4 K[1]} \, dK[1],y(x)\right ]\]
Maple ✓
cpu = 0.081 (sec), leaf count = 29
\[ \left \{ \ln \left ( x \right ) -{\it \_C2}+\int ^{{\frac {y \left ( x \right ) }{{x}^{2}}}}\! \left ( {{\rm e}^{{\it \_f}}}{\it \_C1}+4\,{\it \_f}+2 \right ) ^{-1}{d{\it \_f}}=0 \right \} \] Mathematica raw input
DSolve[x^4*y''[x] == -4*y[x]^2 + x^2*y'[x]*(x + y'[x]),y[x],x]
Mathematica raw output
Solve[C[2] + Integrate[(2 + (2 - E^K[2]^(-2)*C[1])*K[2]^2)/(K[2]*(-4 + (-2 + E^K[2]^(-2)*C[1])*K[2]^2)), {K[2], 1, x}] == Integrate[(x^2*(2 - E^(K[1]/x^2)*C[1]) + 4*K[1])^(-1), {K[1], 1, y[x]}], y[x]]
Maple raw input
dsolve(x^4*diff(diff(y(x),x),x) = x^2*(x+diff(y(x),x))*diff(y(x),x)-4*y(x)^2, y(x),'implicit')
Maple raw output
ln(x)-_C2+Intat(1/(exp(_f)*_C1+4*_f+2),_f = 1/x^2*y(x)) = 0