64.20.16 problem 16

Internal problem ID [13588]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 16
Date solved : Monday, March 31, 2025 at 08:01:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+8 y&=\left \{\begin {array}{cc} 3 & 0<t <2 \pi \\ 0 & 2 \pi <t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.246 (sec). Leaf size: 83

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+8*y(t) = piecewise(0 < t and t < 2*Pi,3,2*Pi < t,0); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 3-{\mathrm e}^{-4 t}+6 \,{\mathrm e}^{-2 t} & t <2 \pi \\ 6-{\mathrm e}^{-8 \pi }+6 \,{\mathrm e}^{-4 \pi } & t =2 \pi \\ 6 \,{\mathrm e}^{-2 t}+6 \,{\mathrm e}^{-2 t +4 \pi }-{\mathrm e}^{-4 t}-3 \,{\mathrm e}^{-4 t +8 \pi } & 2 \pi <t \end {array}\right .\right )}{8} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 94

ode=D[y[t],{t,2}]+6*D[y[t],t]+8*y[t]==Piecewise[{{3,0<t<2*Pi},{0,t>2*Pi}}]; 
ic={y[0]==1,Derivative[1][y][0]==-1}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-4 t} \left (-1+3 e^{2 t}\right ) & t\leq 0 \\ \frac {1}{8} \left (3-e^{-4 t}+6 e^{-2 t}\right ) & 0<t\leq 2 \pi \\ \frac {1}{8} e^{-4 t} \left (-1-3 e^{8 \pi }+6 e^{2 t}+6 e^{2 t+4 \pi }\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((3, (t > 0) & (t < 2*pi)), (0, t > 2*pi)) + 8*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)