63.9.10 problem 1(j)

Internal problem ID [13058]
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 : Wednesday, March 05, 2025 at 09:07:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (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);
 

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

✓ Mathematica. Time used: 1.319 (sec). Leaf size: 164

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]
 

\[ x(t)\to e^{-t/2} \left (\cos \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t-\frac {2 e^{\frac {K[2]}{2}} \left (e^{K[2]} K[2]+5 \sin (2 K[2])\right ) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )}{\sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t\frac {2 e^{\frac {K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) \left (e^{K[1]} K[1]+5 \sin (2 K[1])\right )}{\sqrt {3}}dK[1]+c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.274 (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)
 

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