73.22.11 problem 31.7 (d)

Internal problem ID [15638]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.7 (d)
Date solved : Tuesday, January 28, 2025 at 08:03:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\delta \left (t -2\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: 9.518 (sec). Leaf size: 16

dsolve([diff(y(t),t$2)+16*y(t)=Dirac(t-2),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\operatorname {Heaviside}\left (t -2\right ) \sin \left (4 t -8\right )}{4} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 100

DSolve[{D[y[t],{t,2}]+16*y[t]==DiracDelta[t-2],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -\sin (4 t) \int _1^0\frac {1}{4} \cos (8) \delta (K[2]-2)dK[2]+\sin (4 t) \int _1^t\frac {1}{4} \cos (8) \delta (K[2]-2)dK[2]-\cos (4 t) \int _1^0-\frac {1}{4} \delta (K[1]-2) \sin (8)dK[1]+\cos (4 t) \int _1^t-\frac {1}{4} \delta (K[1]-2) \sin (8)dK[1] \]