Internal problem ID [8360]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 780.
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 }-\frac {x y+y+x \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.581 (sec). Leaf size: 27
dsolve(diff(y(x),x) = (x*y(x)+y(x)+x*(y(x)^2+x^2)^(1/2))/x/(x+1),y(x), singsol=all)
\[ c_{1}+\frac {\sqrt {x^{2}+y \relax (x )^{2}}+y \relax (x )}{x \left (x +1\right )} = 0 \]
✓ Solution by Mathematica
Time used: 22.258 (sec). Leaf size: 58
DSolve[y'[x] == (y[x] + x*y[x] + x*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \tanh (\log (x+1)+c_1)}{\sqrt {\text {sech}^2(\log (x+1)+c_1)}} \\ y(x)\to \frac {x \tanh (\log (x+1)+c_1)}{\sqrt {\text {sech}^2(\log (x+1)+c_1)}} \\ \end{align*}