76.21.14 problem 14 (c)

Internal problem ID [17778]
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 : 14 (c)
Date solved : Tuesday, January 28, 2025 at 11:02:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{4}+y&=\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: 14.286 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+1/4*diff(y(t),t)+y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {8 \,{\mathrm e}^{\frac {1}{8}-\frac {t}{8}} \sqrt {7}\, \operatorname {Heaviside}\left (t -1\right ) \sin \left (\frac {3 \sqrt {7}\, \left (t -1\right )}{8}\right )}{21} \]

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 42

DSolve[{D[y[t],{t,2}]+1/4*D[y[t],t]+y[t]==DiracDelta[t-1],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {8 e^{\frac {1}{8}-\frac {t}{8}} \theta (t-1) \sin \left (\frac {3}{8} \sqrt {7} (t-1)\right )}{3 \sqrt {7}} \]