76.21.1 problem 1

Internal problem ID [17765]
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 : 1
Date solved : Tuesday, January 28, 2025 at 11:02:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=\delta \left (t -\pi \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: 11.754 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 26

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

\[ y(t)\to -e^{-t} \left (e^{\pi } \theta (t-\pi )-1\right ) \sin (t) \]