Differential Equations Dillon & Fadyn Spring 2000

What is a differential equation?

An equation with derivatives in it.

Examples

What was that last line? That is a partial differential equation (P.D.E.) because it has partial derivatives.

Partial Derivatives? When u is a function of two variables, r and t, it has two partial first derivatives, one with respect to r, one with respect to t.

Find the partial derivative of u with respect to t by holding r constant and differentiating as usual.

Example Suppose Then the partial derivative of u with respect to t would be

Notation

Lingo Second partial derivative of u with respect to t Second mixed partial derivatives

Dealing with Differential Equations Determine what the dependent variable is when you are presented with a differential equation. Determine what the independent variable(s) is (are), too.

Example Dependent variable Independent variable

Why? is a function of is dependent on In the equation is the dependent variable, is the independent variable.

Example Dependent variable Independent variable

Why? are all functions of is the dependent variable, is the independent variable. In the equation as evidenced by the right hand side of the equation.

Example Dependent variable Independent variables

Why? is a function ofand as evidenced by the partial derivatives and is the dependent variable, are the independent variables.

Why Does This Matter? We want solutions to differential equations. A solution to a differential equation is a function of the independent variable(s) which can successfully play the role of the dependent variable in the differential equation.

In Other Words The unknown in a differential equation is the dependent variable. It is the thing we want to find. It is the thing whose derivatives appear in the differential equation. It is a function expressed in terms of the independent variable(s).

Example is a solution to because

Example is a solution to because Check that this is true by calculating the derivatives!

Notice In the last example is the dependent variable, but we can call the independent variable or anything we want.

Exercise Rewrite so that the independent variable is clearly

Example is a solution to because Check that for homework!

Homework Read Sections 1.1 and 1.2 in the text Answer the question, ``What is in the text that we didn’t cover in class?’’ On page 8, do problems 1-5, 10, 12, 13, On page 15, do problems 1, 2, 5, 11, 13, 16, 18-22, 28-30, 35, 39