Internal problem ID [7874]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 294.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {x \left (y^{2}+x^{2}-a \right ) y^{\prime }-y \left (y^{2}+x^{2}+a \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 112
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 {1}{\frac {1}{y \relax (x )^{2}}-\frac {1}{-x^{2}+a}} = -\frac {x \sqrt {x^{2}-a}}{\sqrt {c_{1}+\frac {4 a}{x^{2}-a}}}+\frac {x^{2}}{2}-\frac {a}{2} \\ \frac {1}{\frac {1}{y \relax (x )^{2}}-\frac {1}{-x^{2}+a}} = \frac {x \sqrt {x^{2}-a}}{\sqrt {c_{1}+\frac {4 a}{x^{2}-a}}}+\frac {x^{2}}{2}-\frac {a}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.504 (sec). Leaf size: 65
DSolve[x*(y[x]^2+x^2-a)*y'[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*}