68.5.39 problem 43

Internal problem ID [17290]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.3, page 49
Problem number : 43
Date solved : Thursday, October 02, 2025 at 02:00:59 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 33

ode:=diff(y(t),t)+y(t) = piecewise(0 <= t and t < 1,t,1 <= t,0); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 37

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} & t\leq 0 \\ 2 e^{-t} & t>1 \\ t+2 e^{-t}-1 & \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)), (0, t >= 1)) + Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

ValueError : Couldnt solve for initial conditions