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 : Saturday, March 29, 2025 at 10:56:46 PM
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*}

✓ Maple. Time used: 0.290 (sec). Leaf size: 27

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+5*diff(diff(y(t),t),t)+4*y(t) = 1-Heaviside(t-Pi); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 29

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

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

✓ Sympy. Time used: 2.002 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Heaviside(t - pi) + 5*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)) - 1,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\theta \left (t - \pi \right )}{3} - \frac {1}{3}\right ) \cos {\left (t \right )} + \frac {\sin ^{2}{\left (t \right )} \theta \left (t - \pi \right )}{6} + \frac {\cos {\left (2 t \right )}}{12} - \frac {\theta \left (t - \pi \right )}{3} + \frac {1}{4} \]