90.26.6 problem 28

Internal problem ID [25407]
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 379
Problem number : 28
Date solved : Friday, October 03, 2025 at 12:01:13 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y+y^{\prime }&=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 12

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

\[ y = -{\mathrm e}^{\pi -t} \]

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 60

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} -e^{\pi -t} & t>\pi \\ -\frac {1}{2} e^{-t} \left (1+3 e^{\pi }\right ) & t\leq 0 \\ \frac {1}{2} \left (-\cos (t)-3 e^{\pi -t}+\sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.423 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = \begin {cases} \frac {e^{\pi }}{2 e^{t} + 2} + \frac {- \frac {3 e^{\pi }}{2} - 1}{e^{t} + 1} & \text {for}\: t \geq \pi \\\text {NaN} & \text {otherwise} \end {cases} \]