14.19.4 problem 7

Internal problem ID [2697]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.12, Dirac delta function. Excercises page 250
Problem number : 7
Date solved : Monday, January 27, 2025 at 06:11:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&={\mathrm e}^{-t}+3 \delta \left (t -3\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.878 (sec). Leaf size: 30

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t)=exp(-t)+3*Dirac(t-3),y(0) = 0, D(y)(0) = 3],y(t), singsol=all)
 

\[ y = 3 \operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3} \left (t -3\right )+\frac {{\mathrm e}^{-t} t \left (6+t \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 32

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+y[t]==Exp[-t]+3*DiracDelta[t-3],{y[0]==0,Derivative[1][y][0] ==3}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-t} \left (6 e^3 (t-3) \theta (t-3)+t (t+6)\right ) \]