11.5.1 problem 1

Internal problem ID [1506]
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 : 1
Date solved : Monday, January 27, 2025 at 04:58:24 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 )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.357 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 29

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

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