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Solving Systems

General Introduction

If you think back to your course inÌýintroductory algebra, you may remember a progression of topics that went something like this:

• The concept of theÌývariableÌýwas introduced.
• Equations of a single variableÌýwere introduced, along with the idea that they hadÌýsolutionsÌý(which consisted of either real or complex numbers) that satisfied the equation when substituted for the variable.
• Methods of findingÌýexact solutionsÌýwere discussed, starting with very basic methods for solving linear equations, and then leading into more sophisticated methods for solving non-linear equations of various types.
• If you had a really good course, then it was admitted that sometimes it would be necessary toÌýfind the solutions approximately using numerical algorithms.

Doesn't this sequence look familiar when compared to the concepts you have been learning inÌýthisÌýcourse? In fact if you go through the above evolution of topics, and replace each instance of the wordÌý±ð±ç³Ü²¹³Ù¾±´Ç²ÔÌýwith the wordsÌýdifferential equation, and the wordÌývariableÌýwith the wordsÌýunknown function, and you would have a fairly succinct description of our course up through the current laboratory.

So, if you're wondering what might come next in your differential equations course, you might look back to what came next in your algebra course for a hint. You'll recall thatÌýsystems of equationsÌýwere introduced, and that this occurred because in many applications more than a single variable was needed to describe the problems.

The parallel between your algebra course and this differential equations course keeps right on track here! In thousands of physical situations we encounter several functions which are related to one another, and are all dependent on an independent variable, say time. In such instances we developÌýsystems of differential equations, and just like in algebra, in order to solve these systems we need to have the same number of differential equations as we have unknown functions. We also expect the solutions of these systems to consist of sets of functions which when substituted back intoÌýeachÌýof the differential equations in the system renders them all true.

An Introductory Example

As is often the case when introducing a new idea in mathematics, we'll learn the most efficiently if we start with an example.

Let's say that we have two time dependent functions,Ìýx(t) andÌýy(t), and that it has been determined that they are related according to the following system of differential equations:

You can check on paper that a solution set ofÌýx(t) =Ìýe^-Ìýt, andÌýy(t) =Ìýe^3tÌýsatisfies this system. Simply substitute both of these functions (and their derivatives, of course) into both equation (1) and equation (2), and you'll get a true statement in each case. (How weÌýfindÌýthese solutions is another question entirely! You'll address this in the lecture part of your course.)

³¢±ð³Ù'²õÌý³Ü²õ±ðÌýMathematicaÌýto verify that the above solutions are what I've claimed they are.

After you have completed the tutorial come back here for:

I. Linear Systems

Systems and Elimination ->ÌýTwo Masses Spring Systems

Real Distinct Eigenvalues ->ÌýÌýRepeated Eigenvalues ->ÌýÌýSummary

Complex Eigenvalues ->ÌýCenter ->ÌýEllipse Demo

Trace Determinant Plane ->ÌýPHASE PLANE MAIN DEMO

Ìý

Nonhomogeneous Systems ->ÌýMatrix Exponents ->ÌýMechanical Systems

II. Nonlinear Systems

Planar Autonomous Systems ->ÌýStability

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.