Internal problem ID [8630]
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)]`]]
\[ \boxed {x \left (y^{2}+x^{2}-a \right ) y^{\prime }-\left (y^{2}+x^{2}+a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (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 \left (x \right )^{2} \left (-x^{2}+a \right )}{-x^{2}-y \left (x \right )^{2}+a} &= -\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 )}{-x^{2}-y \left (x \right )^{2}+a} &= \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: 0.938 (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*}