Repeated Eigenvalues: How to Solve for a General Solution

[Jamin2112]

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

Homework Statement

Find the general solution of the system of equations.

...

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

Homework Equations

Just watch me solve

The Attempt at a Solution

Assume there's a solution x= $e^rt, 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=$te^rt + #e^rt, where # is another vector with constant entries, and then solve.

 

[Dick]

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

 

[Jamin2112]

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

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

D'oh!

(More than likely, I will some more questions about this assignment. Keep checking into this thread.)

 

[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?

 

[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.