This course seeks to familiarize students with the general ideas of ordinary differential equations and linear systems of such equations, while introducing elementary notions of linear algebra which facilitate the understanding of linear differential equations, either alone or in systems. Emphasis is on first and second order differential equations, those which most frequently occur in applications.
First order differential equations provide a view of the geometric setting of differential equations through the directionfield, and a taste of solution techniques through the two most commonly occurring first order differential equation types: separable and linear. Students should be able to interpret a directionfield plot, and know whether a first order differential equation is separable or linear (in either variable) or neither, and should be able to apply their respective solution algorithms when they are appropriate and the algebraic/calculus operations involved are reasonable, and understand the difference between explicit and implicit solutions.
Students should be be able to solve homogeneous second order constant coefficient differential equations and understand the different kinds of solutions which are possible, including complex characteristic values. [If the current text handles differential equations of any order in this context, emphasis should be on the less exotic possibilities, as in the method of undetermined coefficients, for example.] They should be able to use the method of undetermined coefficients for reasonable driving functions in the nonhomogeneous case and understand the difference between the undetermined coefficients of the technique and the arbitrary constants which appear in the solution. Since the harmonically driven damped harmonic oscillator system is the single most important application of this and connects somewhat to their intuition and prior knowledge from high school physics, it is reasonable that it be used as an example for them. The opportunity to discuss resonance also sets the stage for real applications of eigenvalue analysis (modal analysis) in the linear system case, while the interpretation of the steady state and transient parts of the solution give a physical setting to the mathematical distinction between the homogeneous and nonhomogeneous terms.
In order to appreciate systems of differential equations as well as the nature of higher order linear differential equations, students must acquire some familiarity with matrix algebra and matrix techniques for solving linear systems of equations, including row reduced echelon form with backsubstitution, the determinant (without exaggerating computational techniques), and inverse matrices. Then students should digest the idea of vector spaces, subspaces, span, linear independence and dependence, bases and dimension, with the emphasis on Rn spaces only (and perhaps in passing back to differential equations, mention of the polynomial spaces) and while discussing higher order differential equations, solution spaces of linear homogeneous differential equations as examples, in order to understand the linear aspects of the eigenvalue technique for solving linear (homogeneous) systems of first order ordinary differential equations (with diagonalizable coefficient matrices). It seems reasonable and expected that they should be able to handle complex eigenvalues and eigenvectors in this process. Optionally some exposure to simple nonhomogeneous systems can give them insight into more complicated applications of this process.
A valuable capstone topic is the treatment of a coupled system of harmonic oscillators, where reduction of order shows them how one handles higher order linear systems, the second order case being extremely important for dynamics. Whether or not this is required material for testing, it gives an arena where their intuition about vibrating objects helps them see this eigenvalue technique as a more concrete activity, building on the single harmonic oscillator problem.
Matrix Linear Algebra Note
In addition to supporting the linear foundations of the differential equations
studied in this course, the matrix mathematics covered here also provides many
students their only exposure to a careful treatment of solving linear systems of
equations using row reduction methods. They should thoroughly understand the
reduction process, and the backsubstitution step from the reduced matrix to the
parametrized solution of a linear system of equations A x = b, utilizing
technology for the row echelon reduction process once understood. Going further,
they should also understand the interpretation of a nonhomogeous linear system A x =
b as resolving the problem of finding the most general coefficients of a linear
combination of the columns of the coefficient matrix A
which equals the vector b:
x1Col(A,1) + ... + xn Col(A,n)
= b .
Correspondingly they should understand the
interpretation of the solution of a homogeneous linear system A x = 0 as
finding the most general coefficients of a linear combination of the columns of
A which is the zero vector, i.e., a linear relationship among those vectors when
nonzero. Each free variable (replaced by a parameter in writing down the
solution) in the general solution multiplies a vector of coefficients
representing an independent such linear relationship among those vectors.
All students should learn quickly how to input any combination of differential equations and initial conditions in standard prime derivative notation and how to solve them using the context sensitive menu choice" Solve DE". The course should test the solution algorithms and how to interpret what the solutions mean and what structure they have. Knowing the solution is a good check.
Similarly any matrix is easy to input from the Matrix palette and the context sensitive menu enables the student to find its row reduced form, calculate its determinant and inverse if appropriate (simply a "-1" superscript on the matrix!), and eigenvectors and eigenvalues. The LinearSolveTutor with the Gauss-Jordan reduction choice from the Student[LinearAlgebra] package (available directly from the Tutors menu) should be highlighted in learning the matrix reduction process and in solving linear systems. Matrix multiplication to confirm diagonalization of a square matrix using the matrix of eigenvectors and its inverse is trivial in standard math notation in the Maple input region.