Merced College; Don Power

CALCULUS, CH 9 - STUDY GUIDE

1. Confirm solutions to differential equations (DEs). Examples:

a. Confirm that y = 2sinh (3x) + 5cosh(3x) is a solution of the DE y'' = 9y, with initial value y(0)=5

b. Confirm that y = ln(x) + 3x is a solution of the DE xy' - y =1 - ln(x), with initial value y(1)=3

2. Solve a DE by separation of variables, (with or without initial conditions). Examples:

a. y' = (1 + y²)e^3x with y(0) =

b. e^-y - y'csc²x = 0 with y(0) = ln (5)

3. Solve a first order linear DE by using the integrating factor method (with or without initial conditions): (You will need to start these by rearranging terms)

a. with y(0) = 0

b.

4. Derive the exponential growth/decay equation from the DE y' = ky with initial condition y(0) = y₀

5. Radioactive decay problem

Example: Iodine 131 (¹³¹I) has a half-life of 8 days. If a sample has a mass of 20 grams,

a. Find the mass that remains after t days;

b. Find the mass after 3 days;

c. When will the mass be reduced to 7 g?

6. Newton's Law of Cooling: Set up and solve the differential equation that represents the cooling of a cup of coffee in a 70 degree room if the initial temperature of the coffee is 210 degrees and the temperature after 10 minutes has dropped to 140 degrees. How many minutes will it take for the temperature to reach 100 degrees? Hint: use the initial time and temperature to solve for C, and then use the second data point (10,140) to solve for k.

Return to:  Merced College; Don Power               Updated 11/29/04 by Don Power