ODE
\[ x^4 y''(x)=x \left (x^2+2 y(x)\right ) y'(x)-4 y(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0702465 (sec), leaf count = 83
\[\left \{\left \{y(x)\to \frac {x^2 \left (\left (1-i \sqrt {-c_1-1}\right ) x^{2 i \sqrt {-c_1-1}}+\left (1+i \sqrt {-c_1-1}\right ) c_2\right )}{c_2+x^{2 i \sqrt {-c_1-1}}}\right \}\right \}\]
Maple ✓
cpu = 0.058 (sec), leaf count = 26
\[ \left \{ \ln \left ( x \right ) -{\it \_C2}-{\it \_C1}\,{\it Artanh} \left ( {\frac { \left ( {x}^{2}-y \left ( x \right ) \right ) {\it \_C1}}{{x}^{2}}} \right ) =0 \right \} \] Mathematica raw input
DSolve[x^4*y''[x] == -4*y[x]^2 + x*(x^2 + 2*y[x])*y'[x],y[x],x]
Mathematica raw output
{{y[x] -> (x^2*(x^((2*I)*Sqrt[-1 - C[1]])*(1 - I*Sqrt[-1 - C[1]]) + (1 + I*Sqrt[-1 - C[1]])*C[2]))/(x^((2*I)*Sqrt[-1 - C[1]]) + C[2])}}
Maple raw input
dsolve(x^4*diff(diff(y(x),x),x) = x*(x^2+2*y(x))*diff(y(x),x)-4*y(x)^2, y(x),'implicit')
Maple raw output
ln(x)-_C2-_C1*arctanh((x^2-y(x))/x^2*_C1) = 0