8.12.12 problem 12

Internal problem ID [919]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 12
Date solved : Wednesday, February 05, 2025 at 04:47:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+13 x&=10 \sin \left (5 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: 0.006 (sec). Leaf size: 36

dsolve([diff(x(t),t$2)+6*diff(x(t),t)+13*x(t)=10*sin(5*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {25 \left (2 \cos \left (2 t \right )+5 \sin \left (2 t \right )\right ) {\mathrm e}^{-3 t}}{174}-\frac {25 \cos \left (5 t \right )}{87}-\frac {10 \sin \left (5 t \right )}{87} \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 49

DSolve[{D[x[t],{t,2}]+6*D[x[t],t]+13*x[t]==10*Sin[5*t],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {5}{174} e^{-3 t} \left (25 \sin (2 t)-4 e^{3 t} \sin (5 t)+10 \cos (2 t)-10 e^{3 t} \cos (5 t)\right ) \]