Internal problem ID [2269]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters
Method. page 556
Problem number: Problem 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-4 y-\frac {8}{{\mathrm e}^{2 x}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 46
dsolve(diff(y(x),x$2)-4*y(x)=8/(exp(2*x)+1),y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{-2 x}+c_{1} {\mathrm e}^{2 x}+\left (-{\mathrm e}^{-2 x}+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{2 x}+1\right )-2 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{2 x}-1 \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 47
DSolve[y''[x]-4*y[x]==8/(Exp[2*x]+1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-2 x} \left (-\log \left (e^{2 x}+1\right )+c_2\right )+e^{2 x} \left (2 \tanh ^{-1}\left (2 e^{2 x}+1\right )+c_1\right )-1 \\ \end{align*}