29.23.10 problem 641

Internal problem ID [5232]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 641
Date solved : Monday, January 27, 2025 at 10:37:11 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} x \left (1-x^{2}+y^{2}\right ) y^{\prime }+\left (1+x^{2}-y^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 128

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 {y \left (x \right )^{2} \left (x^{2}-1\right )}{x^{2}-y \left (x \right )^{2}-1} &= -\frac {\sqrt {x -1}\, \sqrt {x +1}\, x}{\sqrt {\frac {c_{1} x^{2}-c_{1} +4}{x^{2}-1}}}-\frac {x^{2}}{2}+\frac {1}{2} \\ \frac {y \left (x \right )^{2} \left (x^{2}-1\right )}{x^{2}-y \left (x \right )^{2}-1} &= \frac {\sqrt {x -1}\, \sqrt {x +1}\, x}{\sqrt {\frac {c_{1} x^{2}-c_{1} +4}{x^{2}-1}}}-\frac {x^{2}}{2}+\frac {1}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.463 (sec). Leaf size: 106

DSolve[x(1-x^2+y[x]^2)D[y[x],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*}