52.8.11 problem 11

Internal problem ID [8375]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 11
Date solved : Monday, January 27, 2025 at 03:49:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=\delta \left (t -\pi \right )+\delta \left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.836 (sec). Leaf size: 61

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+13*y(t)=Dirac(t-Pi)+Dirac(t-3*Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\sin \left (3 t \right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 \pi -2 t}}{3}+\left (\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) {\mathrm e}^{-2 t}-\frac {\sin \left (3 t \right ) \operatorname {Heaviside}\left (t -3 \pi \right ) {\mathrm e}^{-2 t +6 \pi }}{3} \]

✓ Solution by Mathematica

Time used: 0.076 (sec). Leaf size: 59

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==DiracDelta[t-Pi]+DiracDelta[t-3*Pi],{y[0]==1,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\frac {1}{3} e^{-2 t} \left (e^{6 \pi } \theta (t-3 \pi ) \sin (3 t)+e^{2 \pi } \theta (t-\pi ) \sin (3 t)-2 \sin (3 t)-3 \cos (3 t)\right ) \]