Internal problem ID [10441]
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(h).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x-\left (t +2\right ) \sin \left (\pi t \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 106
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=(t+2)*sin(Pi*t),x(t), singsol=all)
\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\frac {\left (\left (-t -2\right ) \pi ^{6}+\left (2 t +7\right ) \pi ^{4}+\left (-2 t -5\right ) \pi ^{2}+t +1\right ) \sin \left (\pi t \right )-\cos \left (\pi t \right ) \pi \left (\left (t +4\right ) \pi ^{4}+\left (-t -6\right ) \pi ^{2}+t +2\right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 114
DSolve[x''[t]+x'[t]+x[t]==(t+2)*Sin[Pi*t],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {-\left (\pi ^2-1\right ) \left (t+\pi ^4 (t+2)-\pi ^2 (t+5)\right ) \sin (\pi t)+\sin (\pi t)-\pi \left (t+\pi ^4 (t+4)-\pi ^2 (t+6)+2\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 ) \\ \end{align*}