1. Spending on Medicare from 2000 to 2005 was projected to rise continuously at an instantaneous rate of 3.7% per year. Find a formula for Medicare spending y as a function of time t in years since 2000. Example: Rising Medical Costs Given: Find: Lecture (26): Ordinary Differential Equations (2 of 2)

2. Solution: Medicare spending is

3. Ex. Find the particular solution to Separable Differential Equations Using (1, 2) we get C = 0

4. Find the particular solution to Exercise: (Waner, Problem #16, Section 7.6) Using (1, 1) we get C = 1 Note that y(1) = 1 > 0

5. Sales  Your monthly sales of Green Tea Ice Cream are falling at an instantaneous rate of 5%/month. If you currently sell 1000 quarters each month, find the differential equation describing your change in sales and then solve it to predict your monthly sales. Given: Find: Exercise: (Waner, Problem #21, Section 7.6)

6. Solution: Apply the initial condition: s(0) = 1000 Monthly sales is

7. Example: Newton’s Law of Cooling Newton’s law of cooling states that a hot object cools at a rate proportional to the difference between its temperature and the temperature of the surrounding environment. If a hot cup of coffee, at 170°F, is left to sit in a room at 70°F, how will the temperature of the coffee change over time? Given: Find:

8. Solution: The Coffee temperature is Apply I.C., H(0) = 170, we get A = 100

9. Heating  Newton’s law of heating is just the same as his law of cooling. The rate of change of temperature is proportional to the difference between the temperature of an object and its surroundings, whether the object is hotter or colder than its surroundings. Suppose that a pie, at 20  F, is put in an oven at 350  F. After 15 minutes its temperature has risen to 80  F. Find the temperature of the pie as a function of time. Exercise: (Waner, Problem #24, Section 7.6) Given: Find:

10. Solution: or

11. Solution: Thus, the temperature of the pie is Apply the initial condition, H(0) = 20, we get Apply the boundary condition, H(15) = 80, we get

12. Exercise: (Waner, Problem #35, Section 7.6) Growth of Tumors  The growth of tumors in animals can be modeled by the Gompertz equation: where y is the size of a tumor, t is time, and a and b are constant that depend on the type of tumor and the units of measurement. a.Solve for y as a function of t. b.If a = 1, b = 10, and y(0) = 5 cm 3 (with t measured in days), find the specific solution and graph it.

13. Solution: