A Brief Introduction to Differential Equations

Michael A. Karls

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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)

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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), y is dependent, x is independent, and k is a

parameter. For (2), u is dependent, x and t are independent, and

there are no parameters.

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How to solve certain differential equations

We now look at how to solve differential equations of the form:

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Case 1: (x,y) = f(x)

In this case we solve by integrating!

We call (5) the general solution to (4). To find a particular solution, we need to specify some initial data such as y(x0)=y0.

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Case 2: (x,y) = f(x)g(y)

In this case, we say the differential equation (3) 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.

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Remark on Case 2:

If g(y0)=0, (6) has a solution of the form y ´ y0, which will be lost in this solution process!

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Example 3

Solve the initial value problem:

Solution: Use separation of variables!

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Solution to Example 3

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Solution to Example 3 (cont.)

Note that y ´ 0 is also a solution to (8). Hence the general solution is:

y = Cekx, with C 2 R. For a particular solution, use (9). 10 = y(0) = Ce0 = C, which implies y = 10ekx.