3.2 Linear Systems of Equations

Question 1

A linear system is homogeneous IF

Answer 1

If the q(t) vector is the zero vector. AKA, there is no q functions in any of the x’, y’.

Question 2

If x1,x2 are homogeneous solns, then

Answer 2

c1x1, c2x2 are also solns of the same equation AKA can be multiplied by constants.

Question 3

Solution matrix is which symbol and in what form?

Answer 3

Capital Phi, and is [x1, x2] where those are the vectors. Or, first column is the first solution, second colm is the second one.

Question 4

How to tell if two solutions are different?

Answer 4

Then the det(PHI) is never 0 for the interval t you are concerned about

Question 5

Fund. Soln. Matrix means:

Answer 5

It is a soln. matrix where the solutions are linearly independent.

Question 6

General Soln of a system of ODEs has the form (in terms of Xp and PHI)

Answer 6

X = Xp + C * PHI
Where C is the C vector and PHI is the fund. soln. matrix.

Question 7

Consider x’ = A(t)x + q(t),
if PHI is any fund. soln. matrix. then, the general solution is:

Answer 7

x(t) = PHI * c + PHI(t) * int( PHI^-1 * q )