76.21.8 problem 8

Internal problem ID [17772]
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 : 8
Date solved : Tuesday, January 28, 2025 at 11:02:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=2 \delta \left (t -\frac {\pi }{4}\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: 13.216 (sec). Leaf size: 16

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

\[ y = -\operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 19

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

\[ y(t)\to \theta (4 t-\pi ) (-\cos (2 t)) \]