A Brief Introduction to Differential Equations

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

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

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(x0)=y0.

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. Remark on Case 2: • If g(y0)=0, (7) has a solution of the form y ´ y0, which will be lost in this solution process!

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

9. Solution to Example 3

10. Solution to Example 3 (cont.) • Note that P ´ 0 is also a solution to (9). Hence the general solution is: • P(t) = Cekt, with C 2 R. For a particular solution, use (10). • P0 = P(0) = Ce0 = C, which implies P(t) = P0ekt.