How Do You Find a Fundamental Set of Solutions Using Jordan Canonical Form?

[mbsl5]

Homework Statement

Find a fundamental set of solutions to the system of differential equations:

y1' = 3*y1 - y2

y2' = y1 + y2

by reducing the problem to Jordan Canonical Form.

Homework Equations

y' = Ay
J = P$$^{-1}$$AP

The Attempt at a Solution

I have found the following:

(1) y' = Ay implies
A =
3 -1
1 1

(2) lamda (eigenvalue) = 2 with multiplicity 2

(3) So I believe
J (JCF) =
2 1
0 2

and P =
1 1
1 0

so that J = P$$^{-1}$$AP

Now I'm stuck. How do I use this to produce the fundamental set of solutions? Any help appreciated.

 

[mighty2000]

Hi there! To find the fundamental set of solutions, we can use the following steps:

1. Find the eigenvalues of A by solving the characteristic equation det(A - lambda*I) = 0. In this case, we have lambda = 2 with multiplicity 2.

2. For each eigenvalue, find the corresponding eigenvector by solving the system (A - lambda*I)x = 0. In this case, we have two eigenvectors: (1, 1) and (-1, 0).

3. Use these eigenvectors to form the matrix P = [x1 x2], where x1 and x2 are the eigenvectors from step 2.

4. Find the inverse of P, P^{-1}.

5. Finally, the fundamental set of solutions is given by e^{lambda*t}*P*P^{-1}, where lambda is the eigenvalue and P*P^{-1} is the inverse of P. In this case, our fundamental set of solutions would be:

y1 = e^{2t}*[1 1]*[1 -1] = e^{2t}

y2 = e^{2t}*[1 1]*[0 0] = 0

Hope this helps! Let me know if you have any further questions. Good luck with your work!