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 : Wednesday, February 05, 2025 at 04:46:42 AM
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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t$2)+4*diff(x(t),t)+5*x(t)=10*cos(3*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 43

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

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