Presentation on theme: "More on Modeling 1.2: Separation of Variables January 18, 2007 HW change: 1.2 #38 is not due this week. Bring your CD-ROM to class Tuesday."— Presentation transcript:
More on Modeling 1.2: Separation of Variables January 18, 2007 HW change: 1.2 #38 is not due this week. Bring your CD-ROM to class Tuesday.
A mystery Here’s a new population model… Find the equilibrium solutions. For what values of P is the population increasing? decreasing? Sketch some solutions to the differential equation. Can you think of a situation in which this model makes sense? Be creative… From last time…
Spread of a rumor (group work) Quantities: (identify as indep var, dep var, or parameter) P = population of city N = people who have heard the rumor t = time k = proportionality constant Answers: 1.dN/dt 2.P - N 3.dN/dt = k(P - N) 4.dN/dt = k(350 - N) (N is in thousands, t could be days, weeks, etc. Your choice of units for t affects the value of k.) What should solutions look like? equilibrium solutions?
mmmm… Chocolate! Quantities: T = temp (degrees F) of hot chocolate at time t t = time in minutes, hours, etc. ( Why doesn’t it matter?) k = proportionality constant Equation: (Should k be positive or negative?)
HUH????? I asked you to look at the statement on p. 22: “So we should never be wrong.” What does that mean? Check this out: Paul says “y 1 (t) = 1 + t is a solution.” Glen says “y 2 (t) = 1 + 2t is a solution.” Bob says “y 3 (t) = 1 is a solution.” Who is right? How can we tell?
Separable Diffy-Q’s Example: Suppose the chocolate started out at 150 o and was 100 o 15 minutes later. How would you solve this initial value problem?