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.

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Presentation transcript:

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 gerbils/month – D 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 ), but the answer turns out to be that I(x) = e  P(x) dx In our example, P(x) = 1/x, I(x) = e  (1/x) dx = x

Another Example Consider the DE xy ' + 3x 3 y = 6x 3 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 = 2x 3 ? – A. 1/x – B. -1/x – C. x – D. e -1/x – E. e 1/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”!