- #1

phy$x

- 3

- 0

ai) Show that (x,y,z) = (1,1,1) is a solution to the following system of equations:

x + y + z = 3

2x + 2y + 2z = 6

3x + 3y +3z = 9

aii) Hence find the general solution of the system

b) Express 2x^2 + 3/(x^2 + 1)^2 in partial fractions

My attempt:

Well ai) was simple and i got that part out with barely any effort.

In aii), i don't even know how to start :( All i know is that the answer is supposed to be:

(x,y,x) = λ(1,0,-1) + μ(0,1,-1) + (1,1,1,)

Sorry i can't offer any attempt...i just really don't know where to start...any help at all will be appreciated here.

With b) i used the matrix method...but that wasnt the approach they were looking for. I was supposed to use the concept of repeated factors:

2x^2 + 3/(x^2 + 1)^2 = Ax + B/x^2 + 1 + CX + D/(x^2+1)^2

(multiply throughout by (x^2 + 1)^2)

2x^2 +3 = (Ax + B)(x^2 + 1) + Cx + D

Let x=0

3 = B + D

D = 3 - B

Let x= 1

5 = (A + B)(2) + C + D

5 = 2A + 2B + C + D

Substituting D = 3 - B

5 = 2A + 2B + C + 3 - B

2 = 2A + B + C

Well this is where I am stuck...any help at all would be a life saver. Thank you in advance.