14.3.21 problem Problem 21

Internal problem ID [3630]
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 21
Date solved : Tuesday, September 30, 2025 at 06:48:46 AM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.080 (sec). Leaf size: 33

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

\[ y = \frac {\left (\left \{\begin {array}{cc} -1+2 x +5 \,{\mathrm e}^{2 x} & x <1 \\ {\mathrm e}^{2 x} \left (5+{\mathrm e}^{-2}\right ) & 1\le x \end {array}\right .\right )}{4} \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{4} \left (2 x+5 e^{2 x}-1\right ) & x\leq 1 \\ \frac {1}{4} e^{2 x-2} \left (1+5 e^2\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.247 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \begin {cases} \frac {x}{2} + \frac {5 e^{2 x}}{4} - \frac {1}{4} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \]