Internal problem ID [8452]
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)]`]]
\[ \boxed {y^{\prime } x -x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y=0} \]
✓ Solution by Maple
Time used: 0.047 (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)
\[ \ln \left (\frac {2 x \left (\sqrt {2 y \left (x \right )^{2}+2 x^{2}}+y \left (x \right )+x \right )}{y \left (x \right )-x}\right )+\frac {\sqrt {2}\, x^{2}}{2}-\ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.405 (sec). Leaf size: 84
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 {x \tanh \left (\frac {x^2+2 c_1}{2 \sqrt {2}}\right ) \left (2+\sqrt {2} \tanh \left (\frac {x^2+2 c_1}{2 \sqrt {2}}\right )\right )}{\sqrt {2}+2 \tanh \left (\frac {x^2+2 c_1}{2 \sqrt {2}}\right )} y(x)\to x \end{align*}