64.20.13 problem 13

Internal problem ID [13585]
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 : 13
Date solved : Monday, March 31, 2025 at 08:01:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 2 & 0<t <4 \\ 0 & 4<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: 0.166 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)+2*y(t) = piecewise(0 < t and t < 4,2,4 < t,0); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} \left ({\mathrm e}^{t}-1\right )^{2} & t \le 4 \\ {\mathrm e}^{t} \left (2 \,{\mathrm e}^{-4}-{\mathrm e}^{-8+t}+{\mathrm e}^{t}-2\right ) & 4<t \end {array}\right . \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 51

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

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

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((2, (t > 0) & (t < 4)), (0, t > 4)) + 2*y(t) - 3*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)