76.21.12 problem 12

Internal problem ID [17776]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 11:02:51 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: 12.030 (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)
 

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

✓ Solution by Mathematica

Time used: 0.103 (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]
 

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