8.12.10 problem 11

Internal problem ID [917]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 12:05:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+5 x&=10 \cos \left (3 t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0\\ x^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 30

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+5*x(t) = 10*cos(3*t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {\left (\cos \left (t \right )-7 \sin \left (t \right )\right ) {\mathrm e}^{-2 t}}{4}-\frac {\cos \left (3 t \right )}{4}+\frac {3 \sin \left (3 t \right )}{4} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 43

ode=D[x[t],{t,2}]+4*D[x[t],t]+5*x[t]==10*Cos[3*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{4} e^{-2 t} \left (-7 \sin (t)+3 e^{2 t} \sin (3 t)+\cos (t)-e^{2 t} \cos (3 t)\right ) \]

✓ Sympy. Time used: 0.272 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(5*x(t) - 10*cos(3*t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \frac {7 \sin {\left (t \right )}}{4} + \frac {\cos {\left (t \right )}}{4}\right ) e^{- 2 t} + \frac {3 \sin {\left (3 t \right )}}{4} - \frac {\cos {\left (3 t \right )}}{4} \]