12.10.6 problem 6

Internal problem ID [1810]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 6
Date solved : Monday, January 27, 2025 at 05:35:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=\frac {4 \,{\mathrm e}^{-x}}{1-{\mathrm e}^{-2 x}} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 50

dsolve(diff(y(x),x$2)-y(x)=4*exp(-x)/(1-exp(-2*x)),y(x), singsol=all)
 

\[ y = \ln \left (1-{\mathrm e}^{-2 x}\right ) {\mathrm e}^{x}+c_2 \,{\mathrm e}^{x}-{\mathrm e}^{-x} \ln \left (-1+{\mathrm e}^{-2 x}\right )+{\mathrm e}^{-x} \ln \left ({\mathrm e}^{-2 x}\right )+{\mathrm e}^{-x} c_1 \]

✓ Solution by Mathematica

Time used: 0.083 (sec). Leaf size: 49

DSolve[D[y[x],{x,2}]-y[x]==4*Exp[-x]/(1-Exp[-2*x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-x} \left (2 e^{2 x} \text {arctanh}\left (1-2 e^{2 x}\right )-\log \left (e^{2 x}-1\right )+c_1 e^{2 x}+c_2\right ) \]