28.3.19 problem 6.54

Internal problem ID [4532]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.54
Date solved : Monday, January 27, 2025 at 09:23:05 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y&=\left (2 t^{2}+t +1\right ) \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-2\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 3.913 (sec). Leaf size: 46

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

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

✓ Solution by Mathematica

Time used: 0.203 (sec). Leaf size: 75

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

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