57.9.8 problem 1(h)

Internal problem ID [14416]
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 : Thursday, October 02, 2025 at 09:36:55 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*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 129

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = (t+2)*sin(Pi*t); 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 122

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.360 (sec). Leaf size: 289

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((-t - 2)*sin(pi*t) + x(t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = - \frac {\pi ^{2} t \sin {\left (\pi t \right )}}{- \pi ^{2} + 1 + \pi ^{4}} + \frac {t \sin {\left (\pi t \right )}}{- \pi ^{2} + 1 + \pi ^{4}} - \frac {\pi t \cos {\left (\pi t \right )}}{- \pi ^{2} + 1 + \pi ^{4}} + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} - \frac {2 \pi ^{6} \sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} - \frac {5 \pi ^{2} \sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} + \frac {\sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} + \frac {7 \pi ^{4} \sin {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} - \frac {4 \pi ^{5} \cos {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} - \frac {2 \pi \cos {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} + \frac {6 \pi ^{3} \cos {\left (\pi t \right )}}{- 2 \pi ^{6} - 2 \pi ^{2} + 1 + 3 \pi ^{4} + \pi ^{8}} \]