General solution of a system of equations and partial fractions

[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 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.

 

[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.

2x^2 + 3/(x^2 + 1)^2

Seems you mean (2x² + 3)/(x² + 1)². Please use parentheses properly and subscript/superscript. Makes expressions much more readable.

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

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.

 

[Ray Vickson]

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 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.

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).

 

[phy$x]

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

Sry about that...

 

[SammyS]

I've been trying to get out this question for a while now:

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

My attempt:

With b) i used the matrix method...but that wasn't 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'm stuck...any help at all would be a life saver. Thank you in advance.

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.

 

[phy$x]

Thanks a lot guys! All help greatly appreciated. :)