Internal problem ID [10442]
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 2.3.1 Nonhomogeneous Equations:
Undetermined Coefficients. Exercises page 110
Problem number: 1(i).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x-4 t -5 \,{\mathrm e}^{-t}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 41
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=4*t+5*exp(-t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +4 t -4+5 \,{\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.892 (sec). Leaf size: 54
DSolve[x''[t]+x'[t]+x[t]==4*t+5*Exp[-t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 4 t+5 e^{-t}+e^{-t/2} \left (c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right )-4 \\ \end{align*}