14.19.3 problem 6

Internal problem ID [2696]
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 : 6
Date solved : Monday, January 27, 2025 at 06:09:32 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=2 \delta \left (t -1\right )-\delta \left (t -2\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: 1.579 (sec). Leaf size: 80

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

\[ y = \frac {4 \sqrt {3}\, \operatorname {Heaviside}\left (-1+t \right ) {\mathrm e}^{-\frac {t}{2}+\frac {1}{2}} \sin \left (\frac {\sqrt {3}\, \left (-1+t \right )}{2}\right )}{3}-\frac {2 \operatorname {Heaviside}\left (t -2\right ) \sqrt {3}\, {\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {\sqrt {3}\, \left (t -2\right )}{2}\right )}{3}+\frac {{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )+3 \cos \left (\frac {\sqrt {3}\, t}{2}\right )\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.233 (sec). Leaf size: 98

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

\[ y(t)\to \frac {1}{3} e^{-t/2} \left (-2 \sqrt {3} e \theta (t-2) \sin \left (\frac {1}{2} \sqrt {3} (t-2)\right )+4 \sqrt {3 e} \theta (t-1) \sin \left (\frac {1}{2} \sqrt {3} (t-1)\right )+\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+3 \cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \]