67.4.32 problem Problem 5(c)

Internal problem ID [14076]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 5(c)
Date solved : Tuesday, January 28, 2025 at 06:13:43 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+29 y&=5 \delta \left (t -\pi \right )-5 \delta \left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.135 (sec). Leaf size: 41

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+29*y(t)=5*Dirac(t-Pi)-5*Dirac(t-2*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\sin \left (5 t \right ) \left (\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 \pi -2 t}+\operatorname {Heaviside}\left (t -2 \pi \right ) {\mathrm e}^{4 \pi -2 t}\right ) \]

✓ Solution by Mathematica

Time used: 0.121 (sec). Leaf size: 93

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+29*y[t]==5*DiracDelta[t-Pi]-5*DiracDelta[t-2*Pi],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^{-2 t} \sin (5 t) \left (\int _1^0-e^{2 K[1]} \cos (5 K[1]) (\delta (K[1]-2 \pi )-\delta (K[1]-\pi ))dK[1]-\int _1^t-e^{2 K[1]} \cos (5 K[1]) (\delta (K[1]-2 \pi )-\delta (K[1]-\pi ))dK[1]\right ) \]