Internal problem ID [3889]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 642.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {x \left (a -x^{2}-y^{2}\right ) y^{\prime }+\left (a +x^{2}+y^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 112
dsolve(x*(a-x^2-y(x)^2)*diff(y(x),x)+(a+x^2+y(x)^2)*y(x) = 0,y(x), singsol=all)
\begin{align*} \frac {1}{\frac {1}{y \left (x \right )^{2}}-\frac {1}{-x^{2}+a}} = -\frac {x \sqrt {x^{2}-a}}{\sqrt {c_{1} +\frac {4 a}{x^{2}-a}}}+\frac {x^{2}}{2}-\frac {a}{2} \frac {1}{\frac {1}{y \left (x \right )^{2}}-\frac {1}{-x^{2}+a}} = \frac {x \sqrt {x^{2}-a}}{\sqrt {c_{1} +\frac {4 a}{x^{2}-a}}}+\frac {x^{2}}{2}-\frac {a}{2} \end{align*}
✓ Solution by Mathematica
Time used: 1.014 (sec). Leaf size: 65
DSolve[x(a-x^2-y[x]^2)y'[x]+(a+x^2+y[x]^2)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (c_1 x-\sqrt {-4 a+\left (4+c_1{}^2\right ) x^2}\right ) y(x)\to \frac {1}{2} \left (\sqrt {-4 a+\left (4+c_1{}^2\right ) x^2}+c_1 x\right ) \end{align*}