Internal problem ID [9134]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 799.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime }-\frac {y \left (-1-{\mathrm e}^{\frac {x +1}{x -1}} x +x^{2} {\mathrm e}^{\frac {x +1}{x -1}} y-x^{2} {\mathrm e}^{\frac {x +1}{x -1}}+y x^{3} {\mathrm e}^{\frac {x +1}{x -1}}\right )}{x}=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 147
dsolve(diff(y(x),x) = y(x)*(-1-x*exp((x+1)/(x-1))+x^2*exp((x+1)/(x-1))*y(x)-x^2*exp((x+1)/(x-1))+x^3*exp((x+1)/(x-1))*y(x))/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\frac {5 \,{\mathrm e}^{\frac {x +1}{x -1}}}{2}} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x +1}{x -1}} x^{2}}{2}} {\mathrm e}^{-2 x \,{\mathrm e}^{\frac {x +1}{x -1}}} {\mathrm e}^{-6 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )}}{x \left (c_{1} +\int -\left (x +1\right ) {\mathrm e}^{\frac {x +1}{x -1}} {\mathrm e}^{\frac {5 \,{\mathrm e}^{\frac {x +1}{x -1}}}{2}} {\mathrm e}^{-\frac {{\mathrm e}^{\frac {x +1}{x -1}} x^{2}}{2}} {\mathrm e}^{-2 x \,{\mathrm e}^{\frac {x +1}{x -1}}} {\mathrm e}^{-6 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )}d x \right )} \]
✓ Solution by Mathematica
Time used: 1.72 (sec). Leaf size: 69
DSolve[y'[x] == (y[x]*(-1 - E^((1 + x)/(-1 + x))*x - E^((1 + x)/(-1 + x))*x^2 + E^((1 + x)/(-1 + x))*x^2*y[x] + E^((1 + x)/(-1 + x))*x^3*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{6 e \operatorname {ExpIntegralEi}\left (\frac {2}{x-1}\right )}}{x \left (e^{6 e \operatorname {ExpIntegralEi}\left (\frac {2}{x-1}\right )}+c_1 e^{\frac {1}{2} e^{\frac {x+1}{x-1}} (x-1) (x+5)}\right )} y(x)\to 0 \end{align*}