29.24.18 problem 680

Internal problem ID [5271]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 680
Date solved : Monday, January 27, 2025 at 10:54:28 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 113

dsolve((a+x^2+y(x)^2)*y(x)*diff(y(x),x) = x*(a-x^2-y(x)^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 6.565 (sec). Leaf size: 149

DSolve[(a+x^2+y[x]^2)*y[x]*D[y[x],x]==x*(a-x^2-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2} \\ y(x)\to \sqrt {-\sqrt {a^2+4 a x^2+4 c_1}-a-x^2} \\ y(x)\to -\sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2} \\ y(x)\to \sqrt {\sqrt {a^2+4 a x^2+4 c_1}-a-x^2} \\ \end{align*}