65.7.9 problem 14.1 (ix)

Internal problem ID [13694]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 14, Inhomogeneous second order linear equations. Exercises page 140
Problem number : 14.1 (ix)
Date solved : Wednesday, March 05, 2025 at 10:12:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }+10 x&={\mathrm e}^{-t} \cos \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)+10*x(t) = exp(-t)*cos(3*t); 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 70

ode=D[x[t],{t,2}]+2*D[x[t],t]+10*x[t]==Exp[-t]*Cos[3*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^{-t} \left (\sin (3 t) \int _1^t\frac {1}{3} \cos ^2(3 K[1])dK[1]+\cos (3 t) \int _1^t-\frac {1}{6} \sin (6 K[2])dK[2]+c_2 \cos (3 t)+c_1 \sin (3 t)\right ) \]

✓ Sympy. Time used: 0.286 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(10*x(t) + 2*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - exp(-t)*cos(3*t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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