Deriving Variation of Parameters for Systems

[rookie404]

1.Homework Statement
We know the derivation of the method of variation of parameters for second order scalar differential. The task is to derive the method of variation of parameters for scalar equations using this approach: first convert the scalar equation into the first order system and then apply the method of variation of parameters for systems.

Hint: Assume that the fundamental set of solutions {y₁(t), y₂(t)} for your scalar equation is known. How would you construct a fundamental matrix for your system from the scalar fundamental set?

Homework Equations

(my work, not in problem statement)
y''+p(t)y'+q(t)y=g(t) <--Second order nonhomogenous scalar
y' = P(t)y + g(t) <--first order nonhomogenous system

The Attempt at a Solution

y''+p(t)y'+q(t)y=g(t)
change of variables...

z(t)= [ z₁(t)
z₂(t) ]

z₁(t) = y(t)
z₂(t) = y'(t)

z'₂(t)+p(t)z₂(t)+q(t)z₁(t)=g(t)
z'₁(t) = y'(t) = z₂(t)
z'₂(t) = y''(t) = -z₂(t)p(t)-z₁(t)q(t)+g(t)

P(t) =
[ 0 1
-q(t) -p(t) ]

G(t) =
[ 0
g(t) ]

y' = P(t)y + g(t) <--now what I have has this form (first order system)

Now I have the variation of parameters for systems derivation in front of me, the end result of which is:

y(t) = \psi(t)u₀ + \psi(t)\int_t_0^t\psi^-1(s)g(s)ds

So I guess the hard part now is finding what the fundamental matrix \psi is. I know the fundamental set is {y₁, y₂}
In the variation of parameters for second order differential equations we make the assumption y₁u'₁ + y₂u'₂ = 0. I am not supposed to make this assumption of this problem.

 

[rookie404]

I know the solution involves creating the fundamental matrix, \phi. After that I should just be able to plug it into the variation of parameters equation I already have at the bottom of my 1st post there. I also know that this fundamental matrix is created using my fundamental set which is {y₁, y₂} for the scalar second order equation. The last thing I know is that I should be using the Wronskian to help me create this matrix, the Wronskian should be the same for the first order system and second order scalar, I believe?
Now where do I go next? Any help is GREATLY appreciated, I have never received help from this forum before, but I have heard good things about it.
Thanks in advance!

 

[rookie404]

Anyone? Anything? Any help would be appreciated.