76.21.6 problem 6

Internal problem ID [17770]
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 : 6
Date solved : Tuesday, January 28, 2025 at 11:02:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}}\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 11.859 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 25

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

\[ y(t)\to \frac {1}{2} (\theta (t-4 \pi ) \sin (2 t)+\cos (2 t)) \]