ODE
\[ x^2 y(x)^2 y''(x)=\left (x^2+y(x)^2\right ) \left (x y'(x)-y(x)\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.06124 (sec), leaf count = 35
\[\left \{\left \{y(x)\to -\frac {x \left (W\left (-e^{-c_2 c_1^2-1} x^{c_1^2}\right )+1\right )}{c_1}\right \}\right \}\]
Maple ✓
cpu = 0.067 (sec), leaf count = 36
\[ \left \{ {\frac {1}{{{\it \_C1}}^{2}}\ln \left ( {\frac {{\it \_C1}\,y \left ( x \right ) +x}{x}} \right ) }-{\frac {y \left ( x \right ) }{{\it \_C1}\,x}}-\ln \left ( x \right ) -{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[x^2*y[x]^2*y''[x] == (x^2 + y[x]^2)*(-y[x] + x*y'[x]),y[x],x]
Mathematica raw output
{{y[x] -> -((x*(1 + ProductLog[-(E^(-1 - C[1]^2*C[2])*x^C[1]^2)]))/C[1])}}
Maple raw input
dsolve(x^2*y(x)^2*diff(diff(y(x),x),x) = (x^2+y(x)^2)*(x*diff(y(x),x)-y(x)), y(x),'implicit')
Maple raw output
1/_C1^2*ln((_C1*y(x)+x)/x)-1/_C1*y(x)/x-ln(x)-_C2 = 0