Internal problem ID [8030]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 450.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {\left (-a^{2}+x^{2}\right ) \left (y^{\prime }\right )^{2}-2 x y^{\prime } y-x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 1.308 (sec). Leaf size: 37
dsolve((-a^2+x^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-x^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} x^{2}-c_{1} a^{2}-\frac {1}{4 c_{1}} \\ y \relax (x ) = c_{1} \sqrt {-a^{2}+x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.433 (sec). Leaf size: 67
DSolve[-x^2 - 2*x*y[x]*y'[x] + (-a^2 + x^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a^2-x^2+c_1{}^2}{2 c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to -\sqrt {(a-x) (a+x)} \\ y(x)\to \sqrt {(a-x) (a+x)} \\ \end{align*}