Internal problem ID [11458]
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]]
\[ \boxed {x^{\prime \prime }+x^{\prime }+x=t \,{\mathrm e}^{-t} \sin \left (\pi t \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 131
dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=t*exp(-t)*sin(Pi*t),x(t), singsol=all)
\[ x \left (t \right ) = -\frac {{\mathrm e}^{-\frac {t}{2}} \left (-c_{1} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \cos \left (\frac {\sqrt {3}\, t}{2}\right )-c_{2} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right )+\left (\left (\pi ^{6} t +\left (-2 t +3\right ) \pi ^{4}+\left (2 t -1\right ) \pi ^{2}-t -1\right ) \sin \left (\pi t \right )-\cos \left (\pi t \right ) \left (\left (t -2\right ) \pi ^{4}+\left (-t +4\right ) \pi ^{2}+t \right ) \pi \right ) {\mathrm e}^{-\frac {t}{2}}\right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 123
DSolve[x''[t]+x'[t]+x[t]==t*Exp[-t]*Sin[Pi*t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to e^{-t} \left (\frac {\left (\pi ^2 (1-2 t)-\pi ^6 t+t+\pi ^4 (2 t-3)+1\right ) \sin (\pi t)+\pi \left (-\pi ^2 (t-4)+\pi ^4 (t-2)+t\right ) \cos (\pi t)}{\left (1-\pi ^2+\pi ^4\right )^2}+c_2 e^{t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]