20.3.21 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 : Monday, January 27, 2025 at 07:47:55 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*}

✓ Solution by Maple

Time used: 0.199 (sec). Leaf size: 31

dsolve([diff(y(x),x)-2*y(x)=piecewise(x<1,1-x,x>=1,0),y(0) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.093 (sec). Leaf size: 45

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

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