Aaron's Life As of Late

This is just a small sampling of what Aaron has been doing in his homework this evening:

Gx=Nxx-brx- a 1 x- a 2 …x- a j G x N x x b r x a 1 x a 2 … x a j (12)

Somewhat more special care must be taken to find the partial fraction expansion. The non-repeated terms are expanded as before, but for the repeated root, an extra fraction is added for each instance of the repeated root:

Gx= β 0 x-br+ β 1 x-br-1+…+ β r - 1 x-b+ α 1 x- a 1 + α 2 x- a 2 +…+ α j x- a j G x β 0 x b r β 1 x b r 1 … β r - 1 x b α 1 x a 1 α 2 x a 2 … α j x a j (13)

All of the alpha constants can be found using the non-repeated roots method above. Finding the beta coefficients (which are due to the repeated root) has the same Heaviside feel to it, except that this time we will add a twist by using the derivative to eliminate some unwanted terms.
Starting off directly with the cover-up method, we can find β 0 β 0 . By multiplying both sides by x-br x b r , we'll get:
x-brGx= β 0 + β 1 x-b+…+ β r - 1 x-br-1+ α 1 x-brx- a 1 + α 2 x-brx- a 2 +…+ α j x-brx- a j x b r G x β 0
β 1 x b … β r - 1 x b r 1 α 1 x b r x a 1
α 2 x b r x a 2 … α j x b r x a j (14)

Now that we have "covered up" the x-br x b r term in the denominator of Gx G x , we plug in x=b x b to each side; this cancels every term on the right-hand side except for β 0 β 0 , leaving the formula
β 0 =x-brGxx=b β 0 x b x b r G x (15)

Just glancing at it makes me want to curl up in the fetal position on the floor and never come out. That is why I'm an English Major :)

*Aaron says the equations didn't copy right, but I can't tell the difference.