11.5.6 problem 6

Internal problem ID [1511]
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 : 6
Date solved : Monday, January 27, 2025 at 04:58:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=2 \delta \left (t -\frac {\pi }{4}\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: 0.334 (sec). Leaf size: 16

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

\[ y = -\operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 28

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

\[ y(t)\to \frac {1}{2} (\sin (2 t)-2 \theta (4 t-\pi ) \cos (2 t)) \]