29.23.11 problem 642

Internal problem ID [5233]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 642
Date solved : Monday, January 27, 2025 at 10:37:15 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 148

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

✓ Solution by Mathematica

Time used: 0.937 (sec). Leaf size: 65

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