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1. (a)

(c)

2. (a) For in ,

(c) For in ,

(e) For in ,

5.

	Polynomial solution

7. (c)

Legendre polynomials

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1. (a) , is a center

(b) , is a stable equilibrium

(c) , is an unstable equilibirum

(d) , is a stable equilibrium

(e) , is an unstable equilibrium

4. (a) Equations:

Graph of :

Phase curve:

(c) Equations:

Graph of :

Phase curve:

(e) Equations:

Graph of :

Phase curve:

5. (a) Graph of (blue) and (red):

Phase curve:

(c) Graph of (blue) and (red):

Phase curve:

7. Graph of :

The period is approximately seconds. The period of the linearized system is seconds. The exact period (see Problem 2 in Section 8.5) is seconds.

9. (a) Equations:

Stationary points: , , and

(b) and are stable equilibrium points and is an unstable equilibrium point.

(c) , , and are all unstable equilibrium points.

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1. centimeters per second per second

3. For ,

and the motion is undamped. For ,

and the motion is underdamped. For ,

and the motion is overdamped.

6. (a) Hint: Show that must satisfy the equation , where .

(b) minutes

(c) miles per hour

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1. (a)

(c)

(e)

(g)

(i)

3. (a) :
:
:

(b) :
:
:

5. (a)

(b)

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1. (a)

(c)

(e)

2. (a)

3. (b)

(c) million

(d) During the year 2022

5. (b) ,

6. (b)

(c)

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1. (a)

(c) , or

(e) , or

(g)

2. (a)

(c) If , domain .
If , domain .

3. (a) The equation of the curve is , where is a constant. This is the equation of an ellipse, which becomes a circle if .

4. (a)

(c)

(e) The slowest population growth occurs when , the most rapid when . The case is called the doomsday model because the population goes to infinity in a finite amount of time.

5. (b)

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1. (a)

(c)

3. (a)

(b) seconds

(c) feet

(d) seconds, rounded to 4 decimal places

4. (a)

(c)

(e)

(g)

5. (a)

(c)

(e)

(g)

7. (b)

(c) Yes, the object appears to be approaching a terminal speed of feet per second.

(d) The solution, , is the terminal speed of the object since this is the speed at which the acceleration is .

9. (b)

(c) million

(d) During the year 2022

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