57.9.19 problem 2(h)

Internal problem ID [14427]
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 : 2(h)
Date solved : Thursday, October 02, 2025 at 09:37:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 51

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

\[ x = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_1 +\frac {\left (-2 t -1\right ) \cos \left (2 t \right )}{8}-\frac {\sin \left (2 t \right ) \left (t -2\right )}{4} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 72

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

\begin{align*} x(t)&\to -\frac {1}{4} (t-2) \sin (2 t)-\frac {1}{8} (2 t+1) \cos (2 t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {7} t}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.168 (sec). Leaf size: 61

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

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