60.1.293 problem 294

Internal problem ID [10307]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 294
Date solved : Monday, January 27, 2025 at 07:05:59 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 148

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

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

✓ Solution by Mathematica

Time used: 0.943 (sec). Leaf size: 65

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