Internal problem ID [10477]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }-x-\frac {{\mathrm e}^{t}}{1+{\mathrm e}^{t}}=0} \]
✓ Solution by Maple
Time used: 0.016 (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: 0.034 (sec). Leaf size: 46
DSolve[x''[t]-x[t]==Exp[t]/(1+Exp[t]),x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} \left (-2 e^t \left (\text {arctanh}\left (2 e^t+1\right )-c_1\right )+e^{-t} \left (\log \left (e^t+1\right )+2 c_2\right )-1\right ) \\ \end{align*}