Internal problem ID [8508]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 928.
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 {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 21
dsolve(diff(y(x),x) = (exp(-y(x)/x)*y(x)*x+exp(-y(x)/x)*y(x)+exp(-y(x)/x)*x^2+exp(-y(x)/x)*x+x)*exp(y(x)/x)/x/(x+1),y(x), singsol=all)
\[ y \relax (x ) = -\ln \left (-\frac {\ln \left (x +1\right )-c_{1}}{x}\right ) x \]
✓ Solution by Mathematica
Time used: 1.886 (sec). Leaf size: 22
DSolve[y'[x] == (E^(y[x]/x)*(x + x/E^(y[x]/x) + x^2/E^(y[x]/x) + y[x]/E^(y[x]/x) + (x*y[x])/E^(y[x]/x)))/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \log \left (\frac {-\log (x+1)+c_1}{x}\right ) \\ \end{align*}