# Test questions

1. Dec 9, 2005

I'm going over old exam questions for the final. I'm not sure what the departament will put on the exams so I'm trying to go over as much as possible, but I having problems figuring certain problems out:

1)
y^(5) + 3y^(4) - 5y''' - 15'' + 4y' + 12y = 0

How do you find the five solutions to the equation and then put it into the 5 dimensional system of first order equations.

2)
Let A be a square matrix, and let Y be a fundamental matrix for the homogeneous linear system x' = Ax.

a) Verify by substituion that e^tA and (YY_0)^-1 are both solutions to the matrix IVP E(t)' = AE(t), E(0) = I, where Y_0 = Y(0).

All I can intepret from this is the e^tA = (YY_0)^-1 so if I can find Y I can slove this...but nothing else...

b) Find e^-A

A = [[1 2],[2,1]]

Any help would be apprecaited, thank you

2. Dec 9, 2005

### hypermorphism

For (1), the standard trick of assuming y=A*ekt gives you a solution basis fast, as all the roots of the characteristic equation are integers.

3. Dec 9, 2005

### George Jones

Staff Emeritus
For 2 b), use eigenvectors and eigenvalues to write -A = S D S^(-1), where D is a diagonal matrix. Note that e^(-A) = e^(S D S^(-1)) = S (e^D) S^(-1).

Regards,
George