52.6.15 problem 65

Internal problem ID [8350]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 65
Date solved : Sunday, March 30, 2025 at 12:52:36 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.179 (sec). Leaf size: 29

ode:=diff(y(t),t)+y(t) = piecewise(0 <= t and t < 1,t,1 <= t,0); 
ic:=y(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \left \{\begin {array}{cc} -1+{\mathrm e}^{-t}+t & t <1 \\ 1+{\mathrm e}^{-1} & t =1 \\ {\mathrm e}^{-t} & 1<t \end {array}\right . \]

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 32

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

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

✗ Sympy

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