76.21.16 problem 15 (a)

Internal problem ID [17780]
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 : 15 (a)
Date solved : Tuesday, January 28, 2025 at 11:02:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{5}+y&=k \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 14.463 (sec). Leaf size: 29

dsolve([diff(y(t),t$2)+1/5*diff(y(t),t)+y(t)=k*Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {10 \,{\mathrm e}^{\frac {1}{10}-\frac {t}{10}} \sqrt {11}\, \operatorname {Heaviside}\left (t -1\right ) k \sin \left (\frac {3 \sqrt {11}\, \left (t -1\right )}{10}\right )}{33} \]

✓ Solution by Mathematica

Time used: 0.078 (sec). Leaf size: 43

DSolve[{D[y[t],{t,2}]+1/5*D[y[t],t]+y[t]==k*DiracDelta[t-1],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {10 k e^{\frac {1-t}{10}} \theta (t-1) \sin \left (\frac {3}{10} \sqrt {11} (t-1)\right )}{3 \sqrt {11}} \]