Internal problem ID [10440]
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(g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x-t \,{\mathrm e}^{-t} \sin \left (\pi t \right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 108
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=t*exp(-t)*sin(Pi*t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {{\mathrm e}^{-t} \left (\left (\pi ^{6} t +\left (-2 t +3\right ) \pi ^{4}+\left (2 t -1\right ) \pi ^{2}-1-t \right ) \sin \left (\pi t \right )-\left (\left (t -2\right ) \pi ^{4}+\left (-t +4\right ) \pi ^{2}+t \right ) \cos \left (\pi t \right ) \pi \right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 117
DSolve[x''[t]+x'[t]+x[t]==t*Exp[-t]*Sin[Pi*t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{-t} \left (\frac {\left (t-\pi ^2 \left (2 t+\pi ^2 \left (\left (\pi ^2-2\right ) t+3\right )-1\right )+1\right ) \sin (\pi t)+\pi \left (-\pi ^2 (t-4)+\pi ^4 (t-2)+t\right ) \cos (\pi t)}{\left (1-\pi ^2+\pi ^4\right )^2}+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 )\right ) \\ \end{align*}