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 = 0.360209 (sec), leaf count = 72
\[\left \{\left \{y(x)\to -\frac {\sqrt {a+e^{-2 c_1 (-\log (x)+c_2)}}}{\sqrt {2} \sqrt {c_1}}\right \},\left \{y(x)\to \frac {\sqrt {a+e^{-2 c_1 (-\log (x)+c_2)}}}{\sqrt {2} \sqrt {c_1}}\right \}\right \}\]
Maple ✓
cpu = 1.387 (sec), leaf count = 48
\[\left [y \left (x \right ) = \frac {\sqrt {\textit {\_C1} \left ({\mathrm e}^{-\textit {\_C2} \textit {\_C1}} x^{\textit {\_C1}}+a \right )}}{\textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {\textit {\_C1} \left ({\mathrm e}^{-\textit {\_C2} \textit {\_C1}} x^{\textit {\_C1}}+a \right )}}{\textit {\_C1}}\right ]\] 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))
Maple raw output
[y(x) = 1/_C1*(_C1*(1/exp(_C2*_C1)*x^_C1+a))^(1/2), y(x) = -1/_C1*(_C1*(1/exp(_C2*_C1)*x^_C1+a))^(1/2)]