90.26.4 problem 26

Internal problem ID [25405]
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 : 26
Date solved : Friday, October 03, 2025 at 12:01:12 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 56

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {1}{9} \left (3 t+e^{-3 t}-1\right ) & 0<t\leq 1 \\ \frac {1}{9} e^{-3 t} \left (1-e^3+3 e^{3 t}\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

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