7.12.15 problem 16

Internal problem ID [397]
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 : 16
Date solved : Wednesday, February 05, 2025 at 03:39:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 70

dsolve(diff(x(t),t$2)+4*diff(x(t),t)+5*x(t)=10*cos(omega*t),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (\omega ^{2}+2 \omega +5\right ) \left (\omega ^{2}-2 \omega +5\right ) \left (\sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 \right ) {\mathrm e}^{-2 t}-10 \omega ^{2} \cos \left (\omega t \right )+40 \omega \sin \left (\omega t \right )+50 \cos \left (\omega t \right )}{\omega ^{4}+6 \omega ^{2}+25} \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 58

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

\[ x(t)\to \frac {40 w \sin (t w)-10 \left (w^2-5\right ) \cos (t w)}{w^4+6 w^2+25}+c_2 e^{-2 t} \cos (t)+c_1 e^{-2 t} \sin (t) \]