Homework Help: Repeated Eigenvalues problem

1. Apr 24, 2010

Jamin2112

This problem, and all the others on this homework assignment, are making me angry.

1. The problem statement, all variables and given/known data

Find the general solution of the system of equations.

.....

x'=[-3 5/2; -5/2 2]x

2. Relevant equations

Just watch me solve

3. The attempt at a solution

Assume there's a solution x= $ert, where I'm denoting a vector with constant entries$.

----> (A-rI)=$----> (A-rI) is singular ----> det(A-rI)=0 ---->(-3-r)(2-r)-(-5/2)(5/2)=0 ----> r= 1/2 ---->(A-(1/2)I)$=(0 0)T

But then I have a problem because the only solution is $=(0 0)T. I'd know how to proceed, were it not for this dilemma. Next I would Assume there's a second solution x=$tert + #ert, where # is another vector with constant entries, and then solve.

2. Apr 24, 2010

Dick

The solution to (-3-r)*(2-r)-(-5/2)*(5/2)=0 is r=(-1/2), isn't it? Not r=1/2.

3. Apr 24, 2010

Jamin2112

r2 + 3r - 2r -6 + 25/4 = 0
r2 + r + 1/4 = 0
(r+1/2)2=0
r= -1/2

D'oh!

4. Apr 24, 2010

Jamin2112

Alright, Dick. Here is that other question I promised you.

I have a problem in the form x'=P(t)x + g(t), and P(t) just happens to be a singular matrix with constant entries. Therefor I can't do a transformation to the form y'= Dy + h(t).

Grrrrr! Now what?

5. Apr 24, 2010

Dick

If P were diagonal, then you could split it into two separate equations. The usual way to do this is to find a basis of R^2 where P is diagonal, then express x and g in terms of that basis and solve. You aren't being very specific here.