ODE
\[ y(x) \left (a-3 x^2-y(x)^2\right ) y'(x)+x \left (a-x^2+y(x)^2\right )=0 \] ODE Classification
[_rational]
Book solution method
Change of Variable, Two new variables
Mathematica ✓
cpu = 0.818839 (sec), leaf count = 39
\[\text {Solve}\left [\frac {1}{2} \left (\frac {a+2 y(x)^2}{x^2+y(x)^2}+\log \left (x^2+y(x)^2\right )\right )=c_1,y(x)\right ]\]
Maple ✓
cpu = 0.339 (sec), leaf count = 122
\[\left [y \left (x \right ) = \frac {\sqrt {-\LambertW \left (-\left (-2 x^{2}+a \right ) \textit {\_C1} \,{\mathrm e}^{2}\right ) \left (x^{2} \LambertW \left (-\left (-2 x^{2}+a \right ) \textit {\_C1} \,{\mathrm e}^{2}\right )-2 x^{2}+a \right )}}{\LambertW \left (-\left (-2 x^{2}+a \right ) \textit {\_C1} \,{\mathrm e}^{2}\right )}, y \left (x \right ) = -\frac {\sqrt {-\LambertW \left (-\left (-2 x^{2}+a \right ) \textit {\_C1} \,{\mathrm e}^{2}\right ) \left (x^{2} \LambertW \left (-\left (-2 x^{2}+a \right ) \textit {\_C1} \,{\mathrm e}^{2}\right )-2 x^{2}+a \right )}}{\LambertW \left (-\left (-2 x^{2}+a \right ) \textit {\_C1} \,{\mathrm e}^{2}\right )}\right ]\] Mathematica raw input
DSolve[x*(a - x^2 + y[x]^2) + y[x]*(a - 3*x^2 - y[x]^2)*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[(Log[x^2 + y[x]^2] + (a + 2*y[x]^2)/(x^2 + y[x]^2))/2 == C[1], y[x]]
Maple raw input
dsolve((a-3*x^2-y(x)^2)*y(x)*diff(y(x),x)+x*(a-x^2+y(x)^2) = 0, y(x))
Maple raw output
[y(x) = 1/LambertW(-(-2*x^2+a)*_C1*exp(2))*(-LambertW(-(-2*x^2+a)*_C1*exp(2))*(x^2*LambertW(-(-2*x^2+a)*_C1*exp(2))-2*x^2+a))^(1/2), y(x) = -1/LambertW(-(-2*x^2+a)*_C1*exp(2))*(-LambertW(-(-2*x^2+a)*_C1*exp(2))*(x^2*LambertW(-(-2*x^2+a)*_C1*exp(2))-2*x^2+a))^(1/2)]