71.15.1 problem 4 (a)

Internal problem ID [14472]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.4, page 265
Problem number : 4 (a)
Date solved : Monday, March 31, 2025 at 12:27:56 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.141 (sec). Leaf size: 22

ode:=diff(y(x),x)+2*y(x) = piecewise(0 <= x and x < 1,2,1 <= x,1); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x),method='laplace');
 

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

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 37

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

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

✗ Sympy

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