Mathematics Homework Help

Section 2.2: 1, 10

Section 2.3: 4

Section 2.4: 2

Section 2.5: 3

Section 2.7: 6

Exercise A

Write the following dierential equations as a rst order autonomous ODE system:

(a) x sin(x_ ) + x = 0

(b) x + t

p

1 + x2 = 3

Exercise B

For r; x0; v0 2 R nd the explicit solutions x(t) of the following dierential equations:

(a) x_ = rx, x(0) = x0,

(b) x_ = rx2, x(0) = x0,

(c) x_ = rx3, x(0) = x0,

(d) x + x_ 2x = 0, x(0) = x0 and x_ (0) = v0:

Analyze the following equations graphically. In each case, sketch the vector field

on the real line, find all the fixed points, classify their stability, and sketch the

graph of x ( t ) for different initial conditions. Then try for a few minutes to obtain

the analytical solution for x ( t ); if you get stuck, don’t try for too long since in several

cases it’s impossible to solve the equation in closed form!

2.2.1 x = 4×2 −16

2.2.10 (Fixed points) For each of (a)–(e), find an equation x f (x) with the

stated properties, or if there are no examples, explain why not. (In all cases, assume

that f ( x ) is a smooth function.)

a) Every real number is a fixed point.

b) Every integer is a fixed point, and there are no others.

c) There are precisely three fixed points, and all of them are stable.

d) There are no fixed points.

e) There are precisely 100 fixed points.

2.3.4 (The Allee effect) For certain species of organisms, the effective growth

rate N N is highest at intermediate N. This is called the Allee effect (Edelstein–

Keshet 1988). For example, imagine that it is too hard to find mates when N is very

small, and there is too much competition for food and other resources when N is

large.

a) Show that N N = r−a(N −b)2 provides an example of the Allee effect, if r, a,

and b satisfy certain constraints, to be determined.

b) Find all the fixed points of the system and classify their stability.

c) Sketch the solutions N ( t ) for different initial conditions.

d) Compare the solutions N ( t ) to those found for the logistic equation. What are

the qualitative differences, if any?

Use linear stability analysis to classify the fixed points of the following systems.

If linear stability analysis fails because f ′(x∗) 0, use a graphical argument to

decide the stability.

2.4.2 x = x(1−x)(2−x)

2.5.3 Consider the equation x = rx+x3 , where r 0 is fixed. Show that x ( t )

in finite time, starting from any initial condition x0 0.

For each of the following vector fields, plot the potential function V ( x ) and identify

all the equilibrium points and their stability.

2.7.6 x = r+x−x3 , for various values of r.

Exercise A

Write the following dierential equations as a rst order autonomous ODE system:

(a) x .. sin(x_ ) + x = 0

(b) x + t

p

1 + x2 = 3

Exercise B

For r; x0; v0 2 R nd the explicit solutions x(t) of the following dierential equations:

(a) x_ = rx, x(0) = x0,

(b) x_ = rx2, x(0) = x0,

(c) x_ = rx3, x(0) = x0,

(d) x + x_ .. 2x = 0, x(0) = x0 and x_ (0) = v0: