12.3.30 problem 38

Internal problem ID [1607]
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 : 38
Date solved : Monday, January 27, 2025 at 05:13:46 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 60

dsolve(diff(y(x),x)-2*y(x)=x*exp(2*x)/(1-y(x)*exp(-2*x)),y(x), singsol=all)
 

\begin{align*} y &= \sqrt {-{\mathrm e}^{-4 x} \left (x^{2}+2 c_1 -1\right )}\, {\mathrm e}^{4 x}+{\mathrm e}^{2 x} \\ y &= -\sqrt {-{\mathrm e}^{-4 x} \left (x^{2}+2 c_1 -1\right )}\, {\mathrm e}^{4 x}+{\mathrm e}^{2 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.725 (sec). Leaf size: 72

DSolve[D[y[x],x]-2*y[x]==x*Exp[2*x]/(1-y[x]*Exp[-2*x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{2 x}-\frac {\sqrt {x^2-1-c_1}}{\sqrt {-e^{-4 x}}} \\ y(x)\to e^{2 x}+\frac {\sqrt {x^2-1-c_1}}{\sqrt {-e^{-4 x}}} \\ \end{align*}