Differential Equations - Solving Systems

Solving Systems

General Introduction

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

  • The concept of the variable was introduced.
  • Equations of a single variable were introduced, along with the idea that they had solutions (which consisted of either real or complex numbers) that satisfied the equation when substituted for the variable.
  • Methods of finding exact solutions were 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 to find the solutions approximately using numerical algorithms.

Doesn't this sequence look familiar when compared to the concepts you have been learning in this course? In fact if you go through the above evolution of topics, and replace each instance of the word equation with the words differential equation, and the word variable with the words unknown 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 that systems of equations were 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 develop systems 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 into each of 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, x(t) and y(t), and that it has been determined that they are related according to the following system of differential equations:

x′ = y - x - e3t (1)
y′ = 3y + 2x - 2e- t (2)

You can check on paper that a solution set of x(t) = e- t, and y(t) = e3t satisfies 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 we find these solutions is another question entirely! You'll address this in the lecture part of your course.)

Let's use Mathematica to 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

Wolfram Technology at Howard University | Howard University ...Systems and Elimination -> Wolfram Technology at Howard University | Howard University ...Two Masses Spring Systems

Wolfram Technology at Howard University | Howard University ...Real Distinct Eigenvalues ->  Wolfram Technology at Howard University | Howard University ...Repeated Eigenvalues ->  Wolfram Technology at Howard University | Howard University ...Summary

Wolfram Technology at Howard University | Howard University ...Complex Eigenvalues -> Wolfram Technology at Howard University | Howard University ...Center -> Wolfram Technology at Howard University | Howard University ...Ellipse Demo

Wolfram Technology at Howard University | Howard University ...Trace Determinant Plane -> Wolfram Technology at Howard University | Howard University ...PHASE PLANE MAIN DEMO

 

Wolfram Technology at Howard University | Howard University ...Nonhomogeneous Systems -> Wolfram Technology at Howard University | Howard University ...Matrix Exponents -> Wolfram Technology at Howard University | Howard University ...Mechanical Systems

II. Nonlinear Systems

Wolfram Technology at Howard University | Howard University ...Planar Autonomous Systems -> Wolfram Technology at Howard University | Howard University ...Stability

Compass 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.