44.6.37 problem 37

Internal problem ID [7181]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 37
Date solved : Sunday, March 30, 2025 at 11:50:24 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+2 y&=\left \{\begin {array}{cc} 1 & 0\le x \le 3 \\ 0 & 3<x \end {array}\right . \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.092 (sec). Leaf size: 32

ode:=diff(y(x),x)+2*y(x) = piecewise(0 <= x and x <= 3,1,3 < x,0); 
ic:=y(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left \{\begin {array}{cc} 0 & x <0 \\ \frac {1}{2}-\frac {{\mathrm e}^{-2 x}}{2} & x <3 \\ \sinh \left (3\right ) {\mathrm e}^{3-2 x} & 3\le x \end {array}\right . \]

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 46

ode=D[y[x],x]+2*y[x]==Piecewise[{ {1,0<=x<=3},{0,x>3}}]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & x\leq 0 \\ \frac {1}{2}-\frac {e^{-2 x}}{2} & 0<x\leq 3 \\ \frac {1}{2} e^{-2 x} \left (-1+e^6\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-Piecewise((1, (x >= 0) & (x <= 3)), (0, x > 3)) + 2*y(x) + Derivative(y(x), x),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)