Internal problem ID [3380]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 641.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {x \left (1-x^{2}+y^{2}\right ) y^{\prime }+\left (1+x^{2}-y^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 114
dsolve(x*(1-x^2+y(x)^2)*diff(y(x),x)+(1+x^2-y(x)^2)*y(x) = 0,y(x), singsol=all)
\begin{align*} \frac {1}{\frac {1}{y \relax (x )^{2}}-\frac {1}{x^{2}-1}} = -\frac {x \sqrt {x -1}\, \sqrt {x +1}}{\sqrt {c_{1}-\frac {2}{x +1}+\frac {2}{x -1}}}-\frac {\left (x -1\right ) \left (x +1\right )}{2} \\ \frac {1}{\frac {1}{y \relax (x )^{2}}-\frac {1}{x^{2}-1}} = \frac {x \sqrt {x -1}\, \sqrt {x +1}}{\sqrt {c_{1}-\frac {2}{x +1}+\frac {2}{x -1}}}-\frac {\left (x -1\right ) \left (x +1\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.121 (sec). Leaf size: 106
DSolve[x(1-x^2+y[x]^2)y'[x]+(1+x^2-y[x]^2)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\sqrt {x^2-4 c_1 x^2+4 c_1{}^2}+x-2 c_1 x}{2 c_1} \\ y(x)\to \frac {\sqrt {x^2-4 c_1 x^2+4 c_1{}^2}+x-2 c_1 x}{2 c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to -x-1 \\ y(x)\to 1-x \\ \end{align*}