28.3.18 problem 6.53

Internal problem ID [4531]
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 : 6.53
Date solved : Monday, January 27, 2025 at 09:22:57 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime }-4 y&=40 t^{2} \operatorname {Heaviside}\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\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 6.535 (sec). Leaf size: 46

dsolve([diff(y(t),t$4)+3*diff(y(t),t$2)-4*y(t)=40*t^2*Heaviside(t-2),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = \left (40 \,{\mathrm e}^{t -2}+8 \,{\mathrm e}^{-t +2}+4 \sin \left (2 t -4\right )-10 t^{2}-15+7 \cos \left (2 t -4\right )\right ) \operatorname {Heaviside}\left (t -2\right ) \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 51

DSolve[{D[y[t],{t,4}]+3*D[y[t],{t,2}]-4*y[t]==40*t^2*UnitStep[t-2],{y[0]==0,Derivative[1][y][0] == 0,Derivative[2][y][0] == 0,Derivative[3][y][0] == 0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -10 t^2+8 e^{2-t}+40 e^{t-2}+7 \cos (4-2 t)-4 \sin (4-2 t)-15 & t>2 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]