ODE
\[ x y(x)^2 y''(x)=\left (a-y(x)^2\right ) y'(x)+x y(x) y'(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 60.1673 (sec), leaf count = 72
\[\left \{\left \{y(x)\to -\frac {\sqrt {a+e^{-2 c_1 \left (c_2-\log (x)\right )}}}{\sqrt {2} \sqrt {c_1}}\right \},\left \{y(x)\to \frac {\sqrt {a+e^{-2 c_1 \left (c_2-\log (x)\right )}}}{\sqrt {2} \sqrt {c_1}}\right \}\right \}\]
Maple ✓
cpu = 0.162 (sec), leaf count = 25
\[ \left \{ \ln \left ( x \right ) -{\frac {\ln \left ( {\it \_C1}\, \left ( y \left ( x \right ) \right ) ^{2}-a \right ) }{{\it \_C1}}}-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[x*y[x]^2*y''[x] == (a - y[x]^2)*y'[x] + x*y[x]*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[a + E^(-2*C[1]*(C[2] - Log[x]))]/(Sqrt[2]*Sqrt[C[1]]))}, {y[x] -> Sqrt[a + E^(-2*C[1]*(C[2] - Log[x]))]/(Sqrt[2]*Sqrt[C[1]])}}
Maple raw input
dsolve(x*y(x)^2*diff(diff(y(x),x),x) = x*y(x)*diff(y(x),x)^2+(a-y(x)^2)*diff(y(x),x), y(x),'implicit')
Maple raw output
ln(x)-1/_C1*ln(_C1*y(x)^2-a)-_C2 = 0