14.3.20 problem Problem 20

Internal problem ID [3629]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.6, First-Order Linear Differential Equations. page 59
Problem number : Problem 20
Date solved : Tuesday, September 30, 2025 at 06:48:45 AM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.100 (sec). Leaf size: 30

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

\[ y = -\frac {\left (\left \{\begin {array}{cc} 1-7 \,{\mathrm e}^{2 x} & x <1 \\ {\mathrm e}^{2 x} \left (-7+{\mathrm e}^{-2}\right ) & 1\le x \end {array}\right .\right )}{2} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} \left (-1+7 e^{2 x}\right ) & x\leq 1 \\ \frac {1}{2} e^{2 x-2} \left (-1+7 e^2\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.222 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \begin {cases} \frac {7 e^{2 x}}{2} - \frac {1}{2} & \text {for}\: x \leq 1 \\\text {NaN} & \text {otherwise} \end {cases} \]