Internal problem ID [1160]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y-\frac {4 \,{\mathrm e}^{-x}}{1-{\mathrm e}^{-2 x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 47
dsolve(diff(y(x),x$2)-y(x)=4*exp(-x)/(1-exp(-2*x)),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-x} c_{2}+c_{1} {\mathrm e}^{x}+\ln \left (1-{\mathrm e}^{-2 x}\right ) {\mathrm e}^{x}+{\mathrm e}^{-x} \left (\ln \left ({\mathrm e}^{-2 x}\right )-\ln \left (-1+{\mathrm e}^{-2 x}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.052 (sec). Leaf size: 49
DSolve[y''[x]-y[x]==4*Exp[-x]/(1-Exp[-2*x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2 e^x \log \left (e^x\right )+(c_1+c_2) \cosh (x)+\sinh (x) \left (2 \log \left (e^x-1\right )+2 \log \left (e^x+1\right )+c_1-c_2\right ) \\ \end{align*}