Internal problem ID [8137]
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: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _Bernoulli]
Solve \begin {gather*} \boxed {x \left (\sqrt {\left (y^{\prime }\right )^{2}+1}+y^{\prime }\right )-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.482 (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)
\[ \frac {c_{1}}{\sqrt {\frac {\left (x^{2}+y \relax (x )^{2}\right )^{2}}{x^{2} y \relax (x )^{2}}}\, \left (-\frac {x^{2}-y \relax (x )^{2}}{2 x y \relax (x )}+\frac {\sqrt {\frac {x^{4}+2 x^{2} y \relax (x )^{2}+y \relax (x )^{4}}{x^{2} y \relax (x )^{2}}}}{2}\right )}+x = 0 \]
✓ Solution by Mathematica
Time used: 0.24 (sec). Leaf size: 35
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*}