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 = 0.79377 (sec), leaf count = 170
\[\text {Solve}\left [\int _1^x\left (\frac {2 y(x)-\left (e^{\frac {y(x)}{K[2]^2}} c_1-2\right ) K[2]^2}{\left (e^{\frac {y(x)}{K[2]^2}} c_1-2\right ) K[2]^3-4 K[2] y(x)}+\int _1^{y(x)}\frac {-4 K[2]^2-2 e^{\frac {K[1]}{K[2]^2}} c_1 \left (K[1]-K[2]^2\right )}{K[2] \left (\left (e^{\frac {K[1]}{K[2]^2}} c_1-2\right ) K[2]^2-4 K[1]\right ){}^2}dK[1]\right )dK[2]+c_2=\int _1^{y(x)}\frac {1}{\left (2-e^{\frac {K[1]}{x^2}} c_1\right ) x^2+4 K[1]}dK[1],y(x)\right ]\]
Maple ✓
cpu = 0.639 (sec), leaf count = 32
\[\left [y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\textit {\_C2} -\left (\int _{}^{\textit {\_Z}}\frac {1}{{\mathrm e}^{\textit {\_f}} \textit {\_C1} +4 \textit {\_f} +2}d \textit {\_f} \right )\right ) x^{2}\right ]\] 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] + Inactive[Integrate][(-((-2 + E^(y[x]/K[2]^2)*C[1])*K[2]^2) + 2*y[x])/((-2 + E^(y[x]/K[2]^2)*C[1])*K[2]^3 - 4*K[2]*y[x]) + Inactive[Integrate][(-4*K[2]^2 - 2*E^(K[1]/K[2]^2)*C[1]*(K[1] - K[2]^2))/(K[2]*(-4*K[1] + (-2 + E^(K[1]/K[2]^2)*C[1])*K[2]^2)^2), {K[1], 1, y[x]}], {K[2], 1, x}] == Inactive[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))
Maple raw output
[y(x) = RootOf(-ln(x)+_C2-Intat(1/(exp(_f)*_C1+4*_f+2),_f = _Z))*x^2]