90.27.9 problem 9

Internal problem ID [25419]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 425
Problem number : 9
Date solved : Friday, October 03, 2025 at 12:01:21 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=\left \{\begin {array}{cc} 0 & 0\le t <1 \\ 6 & 1\le t <3 \\ 0 & 3\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.143 (sec). Leaf size: 59

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+6*y(t) = piecewise(t < 1 and 0 <= t,0,1 <= t and t < 3,6,3 <= t,0); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ 1+2 \,{\mathrm e}^{-3 t +3}-3 \,{\mathrm e}^{2-2 t} & t <3 \\ \left (-2 \,{\mathrm e}^{7}+3 \,{\mathrm e}^{4+t}+2 \,{\mathrm e}-3 \,{\mathrm e}^{t}\right ) {\mathrm e}^{2-3 t} & 3\le t \end {array}\right . \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 6

ode=D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==Piecewise[{ {0, 0<=t<1}, {6,5<=t<3}, {0,t>=3} }]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((0, (t >= 0) & (t < 1)), (6, (t >= 1) & (t < 3)), (0, t >= 3)) + 6*y(t) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)