(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Im supposed to find the nth differential coeff. of [tex]\frac{x^2}{(x-a)(x-b)}[/tex]

2. Relevant equations

3. The attempt at a solution

Using partial fractions, I simplified it to:

[tex]\frac{x^2}{(b-a)}(\frac{1}{x-a}-\frac{1}{x-b})[/tex]

To simplify matters, I assumed [tex]t=\frac{1}{x-a}-\frac{1}{x-b}[/tex].

This expression now becomes [tex]\frac{1}{b-a}x^2t[/tex].

Differentiating wrt x , [tex]\frac{1}{b-a}(2xt+x^2\frac{dt}{dx})[/tex]

I did this two more times, and it seems to be something like this:

[tex]\frac{d^ny}{dx^n}=\frac{1}{b-a}(2nx\frac{d^{n-2}t}{dx^{n-2}}+2nx\frac{d^{n-1}t}{dx^{n-1}}+x^2\frac{d^nt}{dx^n}[/tex]

Now, since [tex]t=\frac{1}{x-a}-\frac{1}{x-b}[/tex], similarly,

[tex]\frac{d^nt}{dx^n}=(-1)^nn!(\frac{1}{(x-a)^{-(n+1)}}-\frac{1}{(x-b)^{-(n+1)}})[/tex]

Substituting this in the above equation, I get another expression which is quite messed up. So far, is what I've done right? And is there another simpler way to do this? It seems to me that I've missed something somewhere which would make this problem a lot simpler, but I just cant find it. I dont think brute force is the only way to do this, and a push in the right direction would really be very welcome.

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# Homework Help: Differential Coefficient

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