MATH 374 Lecture 1

1. MATH 374 Lecture 1 Chapter 1: Definitions, Families of Curves

2. 1.1: Examples of Differential Equations • What is a Differential Equation? • Definition:A differential equation is an equation which involves an unknown function and some of its derivatives. • A common place to find differential equations is in mathematical models.

3. Example 1 • Here are some differential equations:

4. Variables in a Differential Equation • Definition:If an equation involves the derivative of one variable with respect to another, then the former is called a dependent variable and the latter an independent variable. Any variable which is neither independent nor dependent is a parameter.

5. Example 2 • In equations (1) – (3), state which variables are independent, dependent, and parameters. • (1): x independent, y dependent, k and P parameters. • (2): t, x, and y independent, u dependent • (3): t independent, x dependent, m, b, k, and  parameters.

6. A differential equation may also be given in differential form. Consider this equation: 2x(y + 1)dx – y dy = 0 (4) Equation (4) may be written as: 2x(y + 1) – y dy/dx = 0 (5) or 2x(y + 1)dx/dy – y = 0 (6) In (5), x is independent and y is dependent. The roles of x and y are reversed in (6). Thus, if we have an equation in differential form, we will consider either variable to be independent, with the other being dependent. Example 3

7. 1.2: Definitions • Definition:The order of a differential equation is the order of the highest-order derivative present in the equation.

8. Example 4 • The differential equation has order 4. • What about the examples above?

9. nth order ODE • Definition:Any differential equation of the form F(x, y, y’, … , y(n)) = 0 (8) is called an nth order ordinary differential equation.

10. nth order ODE • Often an nth order ODE can be solved explicitly for y(n) in terms of x, y, y’, …, y(n-1) and written in the form y(n) = f(x, y, y’, … , y(n-1)). (9) • In this course, we will always assume we can write an ODE of form (8) in form (9), in one way.

11. nth order ODE • Remark: This is not always the case. For example, the equation x (y’)2 + 4 y’ -6x2 = 0 can be written in two different ways in terms of y’. • See our text, page 20. Use the quadratic formula to solve for y’!

12. Solution of an nth order ODE • Definition:A solution of the nth order ODE (8) is a function  defined on an interval a<x<b such that ’, ’’, … , (n) exist on a<x<b and F(x, (x), ’(x), … , (n)(x)) = 0 for every x in a<x<b.

13. Show that y = 3e4x is a solution of y’’ – 4y’ = 0 on 0<x<5. Solution: y = 3e4x y’ = 12e4x y’’ = 48e4x y, y’, and y’’ are defined on 0<x<5 and y’’ – 4 y’ = 48e4x – 4(12e4x) = 0. Note that there is nothing special about the choice of interval in this example. In fact y = 3e4x is a solution of y’’ – 4 y’ = 0 on any interval a<x<b. For cases like these, we will say the solution exists for all x. Example 5

14. Linear ODE • Definition:An nth order ODE is said to be linear if it can be written in the form: • Which of the above equations are linear?

15. Linear PDE • The notion of linearity can be extended to partial differential equations. • For example, is the general first order linear PDE in two independent variables.