Clicker Question 1

Suppose a population of gerbils starts with 20 individuals and grows at an initial rate of 6% per month. If the maximum capacity is 200, what is the rate of growth (in gerbils/month) when the population is at 150? (Hint: Just use the DE!)

– A. 4.5 gerbils/month– B. 6% gerbils/month– C. 2.25 gerbils/month– D. 1.50 gerbils/month– E. 3.0 gerbils/month

Clicker Question 2

Suppose a population of gerbils starts with 20 individuals and grows at an initial rate of 6% per month. If the maximum capacity is 200, what is the approximate population after 40 months?

– A. 190 gerbils– B. 150 gerbils– C. 110 gerbils– D. 90 gerbils– E. 75 gerbils

Linear Differential Equations (10/24/12)

A differential equation is called linear if the derivative y ' and the dependent variable y each appear linearly only.

That is, the equation can be put in the form y ' + P(x) y = Q(x) .

Note that if Q(x) = 0 then this equation is separable, but otherwise it is not.

Solving Linear DE’s

The idea is to multiply the whole equation by a new function I(x), called the “integrating factor”, which makes the left hand side into an exact derivative resulting from the product rule.

For example, consider the DE y ' + (1/x)y = 2 Try letting I(x) = x . The left hand side is now

the answer to a product rule calculation.

What is the integrating factor?

So, given any linear DE, how can we find the integrating factor I(x)? The derivation is in the text (page 616-17), but the answer turns out to be that

I(x) = eP(x) dx

In our example, P(x) = 1/x , I(x) = e(1/x) dx = x

Another Example

Consider the DE xy ' + 3x3 y = 6x3

What is the integrating factor I(x)? Multiply through by it and note that the left

hand side is exactly (I(x) y) ' . Now solve the DE. What is the specific solution if y = 5 when

x = 0 ?

Clicker Question 3

What is the integrating factor I(x) of the DE x y ' – (1/x)y = 2x3 ?– A. 1/x– B. -1/x– C. x– D. e-1/x

– E. e1/x

Assignment for Monday

Read Section 9.5. In that section, please do Exercises 1-4, 5, 7, 15, 17,

and 19. Extra Credit (due Wed 10/31): 35(a) on page 622.

Use K for the constant of integration rather than C, since c is already being used. Note also that since the object starts “at rest”, we have the initial condition that v=0 when t=0.

Have a great “Study Day”!