63.9.8 problem 1(h)

Internal problem ID [13135]
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)
Date solved : Tuesday, January 28, 2025 at 05:00:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=\left (t +2\right ) \sin \left (\pi t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 129

dsolve(diff(x(t),t$2)+diff(x(t),t)+x(t)=(t+2)*sin(Pi*t),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {t}{2}} c_{1} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+c_{2} {\mathrm e}^{-\frac {t}{2}} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right )+\left (\left (-t -2\right ) \pi ^{6}+\left (2 t +7\right ) \pi ^{4}+\left (-2 t -5\right ) \pi ^{2}+t +1\right ) \sin \left (\pi t \right )-\cos \left (\pi t \right ) \left (\left (4+t \right ) \pi ^{4}+\left (-t -6\right ) \pi ^{2}+t +2\right ) \pi }{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 122

DSolve[D[x[t],{t,2}]+D[x[t],t]+x[t]==(t+2)*Sin[Pi*t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {\left (t-\pi ^6 (t+2)-\pi ^2 (2 t+5)+\pi ^4 (2 t+7)+1\right ) \sin (\pi t)-\pi \left (t+\pi ^4 (t+4)-\pi ^2 (t+6)+2\right ) \cos (\pi t)}{\left (1-\pi ^2+\pi ^4\right )^2}+c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right ) \]