Internal problem ID [3929]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 682.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (-x^{2}+y^{2}+a \right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 122
dsolve((a-3*x^2-y(x)^2)*y(x)*diff(y(x),x)+x*(a-x^2+y(x)^2) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {-\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right ) \left (x^{2} \operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )-2 x^{2}+a \right )}}{\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )} \\ y \left (x \right ) &= -\frac {\sqrt {-\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right ) \left (x^{2} \operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )-2 x^{2}+a \right )}}{\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) {\mathrm e}^{2} c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.336 (sec). Leaf size: 39
DSolve[(a-3*x^2-y[x]^2)*y[x]*y'[x]+x*(a-x^2+y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\frac {a+2 y(x)^2}{x^2+y(x)^2}+\log \left (x^2+y(x)^2\right )\right )=c_1,y(x)\right ] \]