11.5.5 problem 5

Internal problem ID [1510]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:58:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.324 (sec). Leaf size: 15

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

\[ y = \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t -2 \pi \right )+1\right ) \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 16

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

\[ y(t)\to (\theta (t-2 \pi )+1) \sin (t) \]