7.12.16 problem 17

Internal problem ID [398]
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 : 17
Date solved : Wednesday, February 05, 2025 at 03:39:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+45 x&=50 \cos \left (\omega t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 74

dsolve(diff(x(t),t$2)+6*diff(x(t),t)+45*x(t)=50*cos(omega*t),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (\omega ^{2}+12 \omega +45\right ) \left (\omega ^{2}-12 \omega +45\right ) \left (c_1 \cos \left (6 t \right )+c_2 \sin \left (6 t \right )\right ) {\mathrm e}^{-3 t}-50 \omega ^{2} \cos \left (\omega t \right )+300 \omega \sin \left (\omega t \right )+2250 \cos \left (\omega t \right )}{\omega ^{4}-54 \omega ^{2}+2025} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 62

DSolve[D[x[t],{t,2}]+6*D[x[t],t]+45*x[t]==50*Cos[w*t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -\frac {50 \left (\left (w^2-45\right ) \cos (t w)-6 w \sin (t w)\right )}{w^4-54 w^2+2025}+c_2 e^{-3 t} \cos (6 t)+c_1 e^{-3 t} \sin (6 t) \]