Internal problem ID [8892]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 557.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _Bernoulli]
\[ \boxed {x \left (\sqrt {{y^{\prime }}^{2}+1}+y^{\prime }\right )-y=0} \]
✓ Solution by Maple
Time used: 4.438 (sec). Leaf size: 78
dsolve(x*((diff(y(x),x)^2+1)^(1/2)+diff(y(x),x))-y(x)=0,y(x), singsol=all)
\[ \frac {c_{1}}{\sqrt {\frac {\left (y \left (x \right )^{2}+x^{2}\right )^{2}}{y \left (x \right )^{2} x^{2}}}\, \left (-\frac {x^{2}-y \left (x \right )^{2}}{2 y \left (x \right ) x}+\frac {\sqrt {\frac {x^{4}+2 y \left (x \right )^{2} x^{2}+y \left (x \right )^{4}}{y \left (x \right )^{2} x^{2}}}}{2}\right )}+x = 0 \]
✓ Solution by Mathematica
Time used: 0.272 (sec). Leaf size: 37
DSolve[-y[x] + x*(y'[x] + Sqrt[1 + y'[x]^2])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x (x-c_1)} y(x)\to \sqrt {-x (x-c_1)} \end{align*}