12.3.27 problem 35

Internal problem ID [1604]
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
Date solved : Monday, January 27, 2025 at 05:13:36 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y+y^{\prime }&=\frac {2 x \,{\mathrm e}^{-x}}{1+{\mathrm e}^{x} y} \end{align*}

✓ Solution by Maple

Time used: 0.007 (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 &= \left (-1-\sqrt {2 x^{2}-2 c_1 +1}\right ) {\mathrm e}^{-x} \\ y &= \left (-1+\sqrt {2 x^{2}-2 c_1 +1}\right ) {\mathrm e}^{-x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 32.449 (sec). Leaf size: 70

DSolve[D[y[x],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*}