60.1.306 problem 307

Internal problem ID [10320]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 307
Date solved : Monday, January 27, 2025 at 07:08:32 PM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 113

dsolve((y(x)^2+x^2+a)*y(x)*diff(y(x),x)+(y(x)^2+x^2-a)*x = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 6.599 (sec). Leaf size: 149

DSolve[x*(-a + x^2 + y[x]^2) + y[x]*(a + x^2 + y[x]^2)*D[y[x],x]==0,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*}