76.21.4 problem 4

Internal problem ID [17768]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 11:02:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=-20 \delta \left (t -3\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 12.491 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)-y(t)=-20*Dirac(t-3),y(0) = 4, D(y)(0) = 3],y(t), singsol=all)
 

\[ y = -20 \operatorname {Heaviside}\left (t -3\right ) \sinh \left (t -3\right )+4 \cosh \left (t \right )+3 \sinh \left (t \right ) \]

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 47

DSolve[{D[y[t],{t,2}]-y[t]==-20*DiracDelta[t-3],{y[0]==4,Derivative[1][y][0] ==3}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-t-3} \left (e^3 \left (7 e^{2 t}+1\right )-20 \left (e^{2 t}-e^6\right ) \theta (t-3)\right ) \]