7.12.14 problem 15

Internal problem ID [396]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.6 (Forced oscillations and resonance). Problems at page 171
Problem number : 15
Date solved : Wednesday, February 05, 2025 at 03:39:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

dsolve(diff(x(t),t$2)+2*diff(x(t),t)+2*x(t)=2*cos(omega*t),x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 53

DSolve[D[x[t],{t,2}]+2*D[x[t],t]+2*x[t]==2*Cos[w*t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {4 w \sin (t w)-2 \left (w^2-2\right ) \cos (t w)}{w^4+4}+c_2 e^{-t} \cos (t)+c_1 e^{-t} \sin (t) \]