Chapter 2, Problem 1RE

Answer Problems 1–12 without referring back to the text. Fill in the blanks or answer true or false.

1. The linear DE, y′ − ky = A, where k and A are constants, is autonomous. The critical point __________ of the equation is a(n) __________ (attractor or repeller) for k > 0 and a(n) __________ (attractor or repeller) for k < 0.

To determine

To fill: The blanks with appropriate answer.

Answer to Problem 1RE

The critical point $-\frac{A}{k}$ is a repeller for $k>0$ and an attractor for $k<0$.

Explanation of Solution

Given:

The linear differential equation is $y^{\prime }-ky=A$.

Calculation:

The given linear differential equation is,

$y^{\prime }-ky=A$

Simplify the above linear differential equation,

$\begin{array}{l}{y}^{\prime }=A+ky\\ y^{\prime }=k\left(y+\frac{A}{k}\right)\end{array}$

Equate the first derivative to zero to find the critical point.

$\begin{array}{l}k\left(y+\frac{A}{k}\right)=0\\ y+\frac{A}{k}=0\\ y=-\frac{A}{k}\end{array}$

Thus, the critical point $-\frac{A}{k}$ is a repeller for $k>0$ and an attractor for $k<0$.