1. A Brief Introduction to Differential Equations Michael A. Karls

2. 2 What is a differential equation? A differential equation is an equation which involves an unknown function and some of its derivatives. Example 1: (Some differential equations)

3. 3 More Terminology In an equation which involves the derivative of one variable with respect to another variable, the former is called a dependent variable and the latter an independent variable. Any variable which is neither independent nor dependent is a parameter. Example 2: Apply this definition to Example 1.  For (1), t is independent, P is dependent, and k is a parameter.  For (2), t is independent, x is dependent, and m, b, k, and  are parameters.  For (3), x and t are independent, u is dependent, and there are no parameters.

4. 4 How to solve certain differential equations We now look at how to solve differential equations of the form:

5. 5 Case 1:  (x,y) = f(x) In this case we solve by integrating! We call (6) the general solution to (5). To find a particular solution, we need to specify some initial data such as y(x 0 )=y 0.

6. 6 Case 2:  (x,y) = f(x)g(y) In this case, we say the differential equation (4) is separable. To solve, separate variables and integrate! Again, (7) yields a general solution to (6). To find a particular solution, initial data needs to be specified.

7. 7 Remark on Case 2: If g(y 0 )=0, (7) has a solution of the form y ´ y 0, which will be lost in this solution process!

8. 8 Example 3 Solve the initial value problem: Solution: Use separation of variables!

9. 9 Solution to Example 3

10. 10 Solution to Example 3 (cont.) Note that P ´ 0 is also a solution to (9). Hence the general solution is: P(t) = Ce kt, with C 2 R. For a particular solution, use (10). P 0 = P(0) = Ce 0 = C, which implies P(t) = P 0 e kt.