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In formation of differential equation of a given equation what are the things we should eliminate? I have read that if there are n number of arbitrary constants than the order of differential equation so formed will also be n. A question in my textbook says "Obtain the differential equation of all circles of radius a and centre (h,k) that is (x-h)^2+(y-k)^2=a^2." Now I don't know which one h, k or a should be eliminated or all should be eliminated etc.

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all h,k and a will be eliminated because they could assume any value which means they are arbitrary. If one of them has a constant non arbitrary value say a=1 then a would not be eliminated.

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  • $\begingroup$ Thanks that's what I thought but my book eliminated only h and k $\endgroup$ Commented Sep 3, 2016 at 14:05
  • $\begingroup$ But I agree with you $\endgroup$ Commented Sep 3, 2016 at 14:05
  • $\begingroup$ Can you also tell why we need to eliminate arbitrary constants? Thanks $\endgroup$ Commented Sep 3, 2016 at 14:06
  • $\begingroup$ Because we need to make a differential equation of a class of curves that vary only in the value of arbitrary constant. $\endgroup$ Commented Sep 3, 2016 at 14:17
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Only h and k should be eliminated because in this case the radius of the family of circles is given which is a. It is already defined whereas h and k being the centre of the circles can be different keeping a constant. You can also think of it as finding the equation for a family of circles whose radius is known(a in this case) and the only constants that vary for different circles is h and k. Hence they're the arbitrary constants.

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