63.12.9 problem 7

Internal problem ID [13171]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.2 Variation of parameters. Exercises page 124
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 05:11:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }-x&=\frac {{\mathrm e}^{t}}{1+{\mathrm e}^{t}} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 39

dsolve(diff(x(t),t$2)-x(t)=exp(t)/(1+exp(t)),x(t), singsol=all)
 

\[ x \left (t \right ) = c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}+\frac {\left ({\mathrm e}^{-t}-{\mathrm e}^{t}\right ) \ln \left (1+{\mathrm e}^{t}\right )}{2}+\frac {{\mathrm e}^{t} \ln \left ({\mathrm e}^{t}\right )}{2}-\frac {1}{2} \]

✓ Solution by Mathematica

Time used: 0.093 (sec). Leaf size: 51

DSolve[D[x[t],{t,2}]-x[t]==Exp[t]/(1+Exp[t]),x[t],t,IncludeSingularSolutions -> True]
 

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