# General solution of a system of equations and partial fractions

1. Dec 29, 2012

### phy$x I've been trying to get out this question for a while now: 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 dont 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 cant offer any attempt....i just really dont 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 im stuck.....any help at all would be a life saver. Thank you in advance. 2. Dec 29, 2012 ### haruspex Strange question. Clearly all three equations are equivalent, so we can ignore the 2nd and 3rd. Since x+y+z is linear, having found a solution of x+y+z=3, we can add to it any solution of x+y+z=0 and the result will be a solution of the original equation. So the question is essentially asking you to find all solutions of x+y+z=0. Seems you mean (2x2 + 3)/(x2 + 1)2. Please use parentheses properly and subscript/superscript. Makes expressions much more readable. The usual procedure from this point is to separate out each power of x into a different equation. That will give you four equations here. 3. Dec 29, 2012 ### Ray Vickson In (b), do you mean $$2x^2 + \frac{3}{(x^2+1)^2}$$ (which is what you wrote), or do you mean $$\frac{2x^2 + 3}{(x^2 + 1)^2)}?$$ If you meant the former, then what you wrote is perfectly OK, but if you mean the latter, you must use parentheses, like this: (2x^2 + 3)/(x^2+1)^2. Using ASCII and hence things like x^2 is OK, but you must write clearly. Also, where you write Ax + B/x^2 + 1 + CX + D/(x^2+1)^2, you are writing $$A x + \frac{B}{x^2} + 1 + Cx + \frac{D}{(x^2+1)^2}.$$ I hope that is not what you really mean, but again, without parentheses, your expressions are impossible to parse (and, frankly, it takes too much of my time, so I won't even try). 4. Dec 29, 2012 ### phy$x

i meant:
(Ax + B)/(x^2 + 1) + (Cx + D)/((x^2 + 1)^2)

5. Dec 29, 2012

### SammyS

Staff Emeritus
Expanding the product and collecting terms in the equation

2x2 +3 = (Ax + B)(x2 + 1) + Cx + D

gives

$\displaystyle 2x^2+3=Ax^3+Bx^2+(A+C)x+(B+D)\ .$

Now equate coefficients of each power of x.

6. Dec 29, 2012

### phy\$x

Thanks alot guys!!! All help greatly appreciated. :)