I am having a problem understanding,
- Whether to keep a constant (arbitrary or fixed) in the solution of a differential equation
- Figuring which is an arbitrary constant and which a fixed constant
- How many times to differentiate the equation (equal to the number of constants?)
So in the above given questions, in the first one, that is 5th question. We take the original equation $ (x-h) ^2+(y-k) ^2 = a^2 $ and differentiate it twice (not thrice) and get the solution, keeping the constant a in the answer. So I assumed that a must not be an arbitrary constant.
But then in the next one, we take a as an arbitrary constant. Why? (as we differentiate the equation thrice) to ultimately get $ y3=0 $
And if in the last question, r is given so it cannot be an arbitrary constant. Then, the equation doesn't have any arbitrary constants at all. It would become $ x^2 + y^2 - 2ry =0 $ or $ x^2 + y^2 + 2ry =0 $ so it would be a fixed equation and not having any arbitrary constants. So how would we have a differential equation for it? And by my understanding it itself would be the differential equation with degree = 0
I want to know where all am I going wrong.
Thanks for the patience to understand. Let me know if there is some confusion