1. Particular Solutions to Differential Equations Unit 4 Day 2

2. Do Now  Write as many facts as you can relating f ( x ), f '( x ), F ( x ), F ''( x ), and the integral symbol.

3. Ex. 1  What are some solutions of F ' = 3 x – 2?  What could we do to narrow this down to one solution?

4. Ex. 1A  Find the particular solution of F ' = 3 x – 2 that satisfies the initial condition F (2) = 1.

5. Ex. 2  Solve the differential equation with the given initial condition. a) f '( x ) = 4 x + 1, f (3) = 7 b) f '( x ) = 3 x 2 – x, f (-2) = 3 c) f '( x ) = -cos x, f (π) = -2

6. Ex. 3  Solve the differential equation given f ''( x ) = 6 x, f '(3) = 1, and f (-2) = 5.

7. Ex. 3A  Solve the differential equation given f ''( x ) = √ x, f '(0) = 1, and f (1) = 2

8. Ex. 4  The rate of growth of a population of bacteria is given by dP / dt = k √ t, where P the population size, t is the time in days (0 ≤ t ≤ 10), and k is a constant. The initial size of the population is 500. After 1 day, the population has grown to 600. Estimate the population after 9 days.