1. In the autonomous differential equation x=x(1-x), the critical point. Select

Question

1. In the autonomous differential equation x=x(1-x), the critical point. Select the correct answer

a. x=0 is an attractor

b. x=0 is semistable

c. x=1 is an attractor

d. x=1 is a repeller

e. x=1 is semistable

2. The differential equation (x^2 + y^2)y' =xy is

a. linear

b. homogeneous

c. separable

d. exact

e. Bernoulli

3. The autonomous differential equation x=x^2(x-4) has a solution that is

a. increasing everywhere

b. decreasing everywhere

c. increasing if 0<x<4

d. decreasing if x>4

e. increasing if x>4

Explanation / Answer

Let us consider general differenetial equation problems of the form

dfrac{dx}{dt} = f(x)

where the derivative of solutions depends only on x (the dependent variable). Such equations are called autonomous equations. If we think of t as time, the naming comes from the fact that the equation is independent of time.

Let us come back to the cooling coffee problem (see Example 1.3.3). Newton

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