11.4.8 problem 8

Internal problem ID [1502]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 8
Date solved : Monday, January 27, 2025 at 04:58:09 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y&=1-\operatorname {Heaviside}\left (t -\pi \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: 0.336 (sec). Leaf size: 27

dsolve([diff(y(t),t$4)+5*diff(y(t),t$2)+4*y(t)=1-Heaviside(t-Pi),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\left (1+\cos \left (t \right )\right )^{2} \operatorname {Heaviside}\left (t -\pi \right )}{6}+\frac {\left (\cos \left (t \right )-1\right )^{2}}{6} \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 29

DSolve[{D[y[t],{t,4}]+5*D[y[t],{t,2}]+4*y[t]==1-UnitStep[t-Pi],{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {2}{3} \sin ^4\left (\frac {t}{2}\right ) & t\leq \pi \\ -\frac {2 \cos (t)}{3} & \text {True} \\ \end {array} \\ \end {array} \]