Internal problem ID [8140]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 560.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {a y \sqrt {\left (y^{\prime }\right )^{2}+1}-2 x y y^{\prime }+y^{2}-x^{2}=0} \end {gather*}
✗ Solution by Maple
dsolve(a*y(x)*(diff(y(x),x)^2+1)^(1/2)-2*x*y(x)*diff(y(x),x)+y(x)^2-x^2=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 24.761 (sec). Leaf size: 135
DSolve[-x^2 + y[x]^2 - 2*x*y[x]*y'[x] + a*y[x]*Sqrt[1 + y'[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {4 x^2-a^2 (2+c_1 x){}^2}}{\sqrt {-4+a^2 c_1{}^2}} \\ y(x)\to \frac {\sqrt {4 x^2-a^2 (2+c_1 x){}^2}}{\sqrt {-4+a^2 c_1{}^2}} \\ y(x)\to -\frac {\sqrt {-a^2 x^2}}{\sqrt {a^2}} \\ y(x)\to \frac {\sqrt {-a^2 x^2}}{\sqrt {a^2}} \\ \end{align*}