Differential Equations - Solving Differential Equations Analytically

At this point in your differential equations course you should have gotten used to a few of the analytic techniques used to solve first order differential equations. The technique varies somewhat according to the form of the original equation, and you may already have seen forms classified as separable, exact, homogeneous, or linear. The method of solution of each of these forms inevitably involves some type of integration in order to eliminate the unwanted derivative.

For the most part the techniques can become rote exercises, whereby you, the student, classifies the form of the equation, then follows the appropriate recipe to solve it. This kind of repetition sounds like an ideal task for relegating to a computer.

Given this fact, it is surprising how recently personal computer applications were first created to handle the analytic solution of differential equations. (Numerical solutions, on the other hand, were one of the first tasks ever assigned to computers—way back when they were first invented. Indeed, some might even claim that's why they were invented.)

The current state of affairs is that there is a growing set of analytic solvers for personal computers, and Mathematica can claim to be among the best of these. The quality of these programs has improved with each new version of the software, but there are still shortcomings in all of the packages.

In this laboratory you'll be learning to use Mathematica's built in differential equation solver. The command we'll need is DSolve. As usual when starting to use a new command in Mathematica, it's a good idea to ask it to describe the command itself. In this case we do this by issuing the command ?DSolve, followed by the [ENTER] key on the keyboard's numeric keypad, (not the [RETURN] key).

We call the procedure you just completed "checking the syntax" of the command, i.e. we're checking to see exactly what form we're supposed to use when typing the command into Mathematica. (We wouldn't want to get any brackets, braces, commas, etc. in the wrong places.)

Let's now move on to a new page where we can discuss the result that Mathematica just gave you.

After you have completed the tutorial come back here for:

 Picard and Peano Existence and Uniqueness Theorem Examples.

 y'(x)=f(x) equations. Interactive Projectile and Swimmer Problems

Interactive Separable Equations

 Interactive Linear Equations  ->   Application: Mixtures 

Exact Equations and Substitution 

Second Order Linear Equations -> Euler Equations 

Higher Order Linear Equations -> Constant Coefficients -> Complex Coefficients

Nonhomogeneous Equations and Undetermined Coefficients  -> Variation of Parameters

If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.