Internal problem ID [2778]
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]]
\[ \boxed {y^{\prime \prime }-4 y=\frac {8}{{\mathrm e}^{2 x}+1}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
dsolve(diff(y(x),x$2)-4*y(x)=8/(exp(2*x)+1),y(x), singsol=all)
\[ y \left (x \right ) = \left (-{\mathrm e}^{-2 x}+{\mathrm e}^{2 x}\right ) \ln \left ({\mathrm e}^{2 x}+1\right )+\left (c_{1} -2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{2 x}+{\mathrm e}^{-2 x} c_{2} -1 \]
✓ Solution by Mathematica
Time used: 0.086 (sec). Leaf size: 56
DSolve[y''[x]-4*y[x]==8/(Exp[2*x]+1),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-2 x} \left (2 e^{4 x} \text {arctanh}\left (2 e^{2 x}+1\right )-e^{2 x}-\log \left (e^{2 x}+1\right )+c_1 e^{4 x}+c_2\right ) \]