57.9.10 problem 1(j)

Internal problem ID [14418]
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(j)
Date solved : Thursday, October 02, 2025 at 09:36:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=5 \sin \left (2 t \right )+{\mathrm e}^{t} t \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 50

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = 5*sin(2*t)+t*exp(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 -\frac {10 \cos \left (2 t \right )}{13}-\frac {15 \sin \left (2 t \right )}{13}+\frac {\left (t -1\right ) {\mathrm e}^{t}}{3} \]

✓ Mathematica. Time used: 1.746 (sec). Leaf size: 83

ode=D[x[t],{t,2}]+D[x[t],t]+x[t]==5*Sin[2*t]+t*Exp[t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{39} \left (-13 e^t+13 e^t t+30 \sin ^2(t)-30 \cos ^2(t)-90 \sin (t) \cos (t)+39 c_2 e^{-t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+39 c_1 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t*exp(t) + x(t) - 5*sin(2*t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\left (t - 1\right ) e^{t}}{3} + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} - \frac {15 \sin {\left (2 t \right )}}{13} - \frac {10 \cos {\left (2 t \right )}}{13} \]