11.5.10 problem 10(c)

Internal problem ID [1515]
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 : 10(c)
Date solved : Monday, January 27, 2025 at 04:58:36 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: 0.449 (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)
 

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

✓ Solution by Mathematica

Time used: 0.070 (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]
 

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