Bill Goodwine's Courses

Unless otherwise indicated, all problems are from the course text, Elementary Differential Equations and Boundary Value Problems, by Boyce and DiPrima, 8th Edition. You may use a computer to find the roots of any polynomial with degree greater than two.

1. Section 7.8, numbers 7 and 10.
2. Find the general solution to
   • 
   where
   • 
3. Find the general solution to
   • 
   where
   • 
4. Find the general solution to
   • 
   where
   • 
5. Find the general solution to
   • 
   where
   • 
6. For the case where an eigenvalue has an algegraic multiplicity of m, then if
   • 
   show that
   • 
   satisfies
   • 
   by substituting it into the differential equation and equating powers of the independent variable.
7. Write a computer program in C, C++ or Fortran to determine an approximate numerical solution to
   • 
   Assume that all the initial conditions are equal to zero. Submit your computer code and a plot comparing your numerical solution to the exact solution you determined for this problem in the previous homework.

For the two textbook problems, we get a root (r) that is repeated twice. (m=2) Following our procedure, we would have matrix A as:

A = [x1 x2; x3 x4] where x1, x2, x3, and x4 are integers.

We now must find A*A. If A*A = [0 0; 0 0], is the generalized eigenvector always [1; 0]? From what I can see, it could be any integer, and therefore, there could be infinite solutions? Or can we not apply our method to the 2x2.

Thanks in advance.

awulz wrote:For the two textbook problems, we get a root (r) that is repeated twice. (m=2) Following our procedure, we would have matrix A as:

A = [x1 x2; x3 x4] where x1, x2, x3, and x4 are integers.

We now must find A*A. If A*A = [0 0; 0 0], is the generalized eigenvector always [1; 0]? From what I can see, it could be any integer, and therefore, there could be infinite solutions? Or can we not apply our method to the 2x2.

Thanks in advance.

Do you mean (A - lambda I)^2 = 0? If so, then the reduced matrix is just all zeros and the two solutions are [1 0] and [0 1].

There are an infinite number of solutions and there always will be. In the case of just a regular eigenvalue, you can scale it arbitrarily, so there are an infinite number of solutions. In the case were the dimension of the null space of the generalized eigenspace is greater than one, you need to find a basis. For matrices of real numbers, there will always be an infinite number of choices for this. The procedure that I gave in class is just one way to find a set of solutions.

In question 3 of the homework, we have two cases of repeated eigenvalues. Lamba = 4 repeats twice and lamba =2 repeats three times. I obtained eigenvectors of [0; 0; 0; 1; 0] and [0; 0; 0; 0; 1] for the lamba = 4 case, yet when I begin the lamba = 2 case, I obtain eigenvectors of [0; 0; 1; 0; 0], [0; 0; 0; 1; 0], and [0; 0; 0; 0; 1]. Is it possible to 're-use' the two eigenvectors that appeared in the first eigenvalue case or does this make them not linearly independent of each other because they are occurring more than once?

acurrie wrote:In question 3 of the homework, we have two cases of repeated eigenvalues. Lamba = 4 repeats twice and lamba =2 repeats three times. I obtained eigenvectors of [0; 0; 0; 1; 0] and [0; 0; 0; 0; 1] for the lamba = 4 case, yet when I begin the lamba = 2 case, I obtain eigenvectors of [0; 0; 1; 0; 0], [0; 0; 0; 1; 0], and [0; 0; 0; 0; 1]. Is it possible to 're-use' the two eigenvectors that appeared in the first eigenvalue case or does this make them not linearly independent of each other because they are occurring more than once?

I haven't checked by hand before typing this, but I strongly suspect that your generalized eigenvectors for lambda=4 should be [1;0;0;0;0] and [0;1;0;0;0]. Your generalized eigenvectors for lambda=2 look o.k. to me.

In problem 6 of the homework it says to do the proof by equating the powers of the independent variables. I am unsure about how to do this and I was wondering if you might be able to provide an example in class or explain the procedure. Thank you!

rpaietta wrote:In problem 6 of the homework it says to do the proof by equating the powers of the independent variables. I am unsure about how to do this and I was wondering if you might be able to provide an example in class or explain the procedure. Thank you!

The hint I will give you is that you did this a lot when you learned about the method of undetermined coefficients, especially when the inhomogeneous term was a polynomial.

Someone asked me:

I know we are allowed to use Matlab to compute the eigenvalues of the 5x5 matrices in this weeks homework. However, for problems 2 through 5 on hw 4 are we allowed to use a computer to compute A^2, A^3, multiply 5x1 vectors into 5x5 matrices, or reduce 5x5 matrices to reduced row echelon form when plugging in to the long formula derived in class for the multiple eigenvalue cases to compute the general solution of the system? Thank you.

Yes, but 1) it will take longer to type them in and get it all right than to just do it by hand and 2) you won't have a computer available on the test so using a computer for this on a homework probably will be detrimental to a good performance on an exam.

I don't have the Boyce and DiPrima Book... should i try to borrow it for the homework or disregard question 1?

Luke Nakatsukasa wrote:I don't have the Boyce and DiPrima Book... should i try to borrow it for the homework or disregard question 1?

Yes, skip the first one. If you get a chance, email me exactly what Jason assigned and I'll put it together in a new homework up at the top.