11.5.8 problem 8

Internal problem ID [1513]
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 : 8
Date solved : Monday, January 27, 2025 at 04:58:32 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.347 (sec). Leaf size: 21

dsolve([diff(y(t),t$4)-y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\operatorname {Heaviside}\left (-1+t \right ) \left (\sin \left (-1+t \right )-\sinh \left (-1+t \right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 44

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

\[ y(t)\to \frac {1}{4} e^{-t-1} \theta (t-1) \left (e^{2 t}+2 e^{t+1} \sin (1-t)-e^2\right ) \]