Internal problem ID [954]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number: 35.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime }+y-\frac {2 x \,{\mathrm e}^{-x}}{1+{\mathrm e}^{x} y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 47
dsolve(diff(y(x),x)+y(x)=(2*x*exp(-x))/(1+y(x)*exp(x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (-1-\sqrt {2 x^{2}-2 c_{1}+1}\right ) {\mathrm e}^{-x} \\ y \relax (x ) = \left (-1+\sqrt {2 x^{2}-2 c_{1}+1}\right ) {\mathrm e}^{-x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 32.54 (sec). Leaf size: 70
DSolve[y'[x]+y[x]==(2*x*Exp[-x])/(1+y[x]*Exp[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{-2 x} \left (e^x+\sqrt {e^{2 x} \left (2 x^2+1+c_1\right )}\right ) \\ y(x)\to e^{-2 x} \left (-e^x+\sqrt {e^{2 x} \left (2 x^2+1+c_1\right )}\right ) \\ \end{align*}