Internal problem ID [7696]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 115.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime } x -x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.072 (sec). Leaf size: 49
dsolve(x*diff(y(x),x) - x*(y(x)-x)*sqrt(y(x)^2 + x^2) - y(x)=0,y(x), singsol=all)
\[ \ln \left (\frac {2 x \left (\sqrt {2 x^{2}+2 y \relax (x )^{2}}+y \relax (x )+x \right )}{-x +y \relax (x )}\right )+\frac {\sqrt {2}\, x^{2}}{2}-\ln \relax (x )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.997 (sec). Leaf size: 71
DSolve[x*y'[x] - x*(y[x]-x)*Sqrt[y[x]^2 + x^2] - y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} x \left (\sqrt {2} \tanh \left (\frac {x^2+2 c_1}{2 \sqrt {2}}\right )-\frac {1}{1+\sqrt {2} \tanh \left (\frac {x^2+2 c_1}{2 \sqrt {2}}\right )}+1\right ) \\ y(x)\to x \\ \end{align*}