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At this point in your differential equations course you should have gotten used to a few of theÌýanalyticÌýtechniques 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 asÌýseparable, exact, homogeneous, orÌýlinear. 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 theÌýanalyticÌýsolution of differential equations. (NumericalÌýsolutions, 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'sÌýwhyÌýthey were invented.)

The current state of affairs is that there is a growing set of analytic solvers for personal computers, andÌýMathematicaÌýcan 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 useÌýMathematica's built in differential equation solver. The command we'll need isÌýDSolve. As usual when starting to use a new command inÌýMathematica, it's a good idea to ask it to describe the command itself. In this case we do this by issuing the commandÌý?DSolve, followed by the [ENTER] key on the keyboard's numeric keypad, (notÌýthe [RETURN] key).

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

Let's nowÌýmove onÌýto a new page where we can discuss the result thatÌýMathematicaÌýjust gave you.

After you have completed the tutorial come back here for:

ÌýPicard and Peano Existence and Uniqueness Theorem Examples.

Ìýy'(x)=f(x) equations.ÌýInteractive Projectile and Swimmer Problems

Ìý

Interactive Separable Equations

ÌýInteractive Linear EquationsÌý ->Ìý ÌýApplication: MixturesÌý

Exact Equations and SubstitutionÌý

Second Order Linear Equations ->ÌýEuler EquationsÌý

Ìý

Higher Order Linear Equations ->ÌýConstant CoefficientsÌý->ÌýComplex Coefficients

Nonhomogeneous Equations and Undetermined CoefficientsÌýÌý->ÌýVariation 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.