58.5.29 problem 29

Internal problem ID [14623]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number : 29
Date solved : Thursday, October 02, 2025 at 09:44:37 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+y&=\left \{\begin {array}{cc} {\mathrm e}^{-x} & 0\le x <2 \\ {\mathrm e}^{-2} & 2\le x \end {array}\right . \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.083 (sec). Leaf size: 35

ode:=diff(y(x),x)+y(x) = piecewise(0 <= x and x < 2,exp(-x),2 <= x,exp(-2)); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 40

ode=D[y[x],x]+y[x]==Piecewise[{{Exp[-x],0<=x<2},{Exp[-2],x>=2}}]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-x} & x\leq 0 \\ \frac {1}{e^2}+2 e^{-x} & x>2 \\ e^{-x} (x+1) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-Piecewise((exp(-x), (x >= 0) & (x < 2)), (exp(-2), x >= 2)) + y(x) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)