09.11.2016
Dear Student,
I hope that your work this past week has allowed you to catch up after joining my class a bit late.
Week 4 begins tomorrow, Sept. 12 and covers only two lectures Lectures 9 and 10 and the associated homework
Our class schedule for Week 4 prescribes that you should complete Lectures 9 and 10 and the associated homework by Sunday, September 18:
Week 4 - Content Summary for Lectures 9 and 10:
Lecture 10 begins with some applications of linear systems of second order linear differential equations to the analysis of coupled mechanical vibrations. We show how the fundamental modes of vibration of such a system are determined by the eigenvalues and corresponding eigenvectors of the matrix of the system. You will see the eigenvalues and eigenvectors of a matrix are not simply mathematical items that you can calculate as we have done in earlier lectures but that, more importantly, are descriptive of the fundamental modes of vibration of the physical system modeled by the system of differential equations.
The second part of Lecture 10 and Lecture 11 present four different methods for finding a particular solution yp(t) of a non-homogeneous system y’(t) = A y(t) +g(t) where A is a constant matrix. This is important because the complete set of solutions of a non-homogeneous linear system:
y’((t) = A y(t) +g(t)
is given by y(t) = yh(t) + yp(t) where yh(t) is the complete set of solutions of the associated homogeneous system y'(t) = A y(t) and yp(t) is any solution of the given non-homogeneous system. This is an important consequence of the linearity of the system.
We develop four different methods for finding particular solutions of non-homogeneous first order differential equations. Of these four methods, the Method of Undetermined Coefficients and the Method of Variation of Parameters are the most useful in practice.
Additional Comments on the content of Lectures 9, 10 and 11:
Although the term “fundamental matrix” is mentioned in Lecture 9 and again in Lecture 11, the meaning of this term is not precisely defined and a good method for computing it is not given. Here is a summary of the content that is not adequately covered in the course lectures.:
1. What is a "fundamental matrix" of a system of first order linear differential equations?
Definition of the Fundamental Matrix of a first order linear system:
Suppose y’(t) = A y(t) is a homogeneous system of first order linear differential equations where A is an n by n matrix.
If you apply the Exponential Matrix Method to solve the system, then, in at most n steps, you arrive at n linearly independent solutions: y1(t), …,yk(t) where k is at most n. The resulting n by k matrix is a fundamental matrix F(t) = [y1(t), …,yk(t) ] for the given system.
2. In practice, how do you compute a fundamental matrix for given system and why is it useful? Fundamental matrices for a give system are not uniquely determined by the system but rather depend on how the corresponding eigenvectors are selected and also how the non-eigenvector solutions are constructed. However, for any Fundamental matrix F(t) of the system, it can be shown that the following matrix equation holds:
(*) e^(A t) = F(t) . InverseF(0)
This fact gives a useful tool for computing e^(A t) without summing the infinite series of matrices used to define it. This is in fact, the way in which programs like Mathematica compute e^(A t). However, this equation is not adequately explained in the lectures but it is used in the homework problems in Lectures 9 and !0.
Comment: The equation e^(A t) = F(t) . InverseF(0) the matrix version of the following fact about the solution of the first order DE: y’(t) = a y(t) where a is a number. Recall that to solve such an equation, you would proceed as follows:
a = y’(t) / y(t)) = d/dt [Log(y(t)]. Next, ntegrate-by-parts with respect to t, to get
y(t) = e^(a t + c) = e^(a 0) = e^(a t) where c is the constant of integration.
Replace t by 0 in this equation to get y(t) = e^(a 0) e^c.
Therefore, y(t) = e^(a t) so e^((a t) / y(0),
which is just the scalar version of (*)
I hope that you find these explanatory remarks helpful as you study the lectures and homework problems in Lectures 9 and 10 and the associated homework for Week 4!
Please contact me if you have any questions.
Have a pleasant and productive week!
Tony
Cell: 217-840 2871