73.22.19 problem 31.7 (L)

Internal problem ID [15646]
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 (L)
Date solved : Tuesday, January 28, 2025 at 08:03:34 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-16 y&=\delta \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.338 (sec). Leaf size: 17

dsolve([diff(y(t),t$4)-16*y(t)=Dirac(t),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\sin \left (2 t \right )}{16}+\frac {\sinh \left (2 t \right )}{16} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 193

DSolve[{D[y[t],{t,4}]-16*y[t]==DiracDelta[t],{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{128} e^{-2 t} \left (-128 e^{4 t} \int _1^0\frac {\delta (K[1])}{32}dK[1]+128 e^{4 t} \int _1^t\frac {\delta (K[1])}{32}dK[1]+128 \int _1^t-\frac {\delta (K[2])}{32}dK[2]-128 e^{2 t} \sin (2 t) \int _1^0-\frac {\delta (K[3])}{16}dK[3]+128 e^{2 t} \sin (2 t) \int _1^t-\frac {\delta (K[3])}{16}dK[3]-128 \int _1^0-\frac {\delta (K[2])}{32}dK[2]-\delta ''(0)+\delta (0) e^{4 t}-2 \delta (0) e^{2 t} \cos (2 t)+\delta (0)+e^{4 t} \delta ''(0)-2 e^{2 t} \delta ''(0) \sin (2 t)\right ) \]