Hi all,

Working though this problem, I have reached a proof I am not sure of the method, any opinions much appreciated!

Q:

where ; then satisfies :

where

------ This part is fine with some partial differentiation, chain rule and substitution, I can get the proof.

2nd proof:

show that is independant of if

------ This proof is also fine with some partial differentiation and change of variable

ending with

3rd proof:

further show when where is an arbitary constant....

Now given the second proof, is it possible to integrate the entire expression

where w r t : ?

reducing the orders of the differential terms? I suspect this is not possible since one of the terms contains a ..... if we cannot directly integrate to reduce the power, is it possible to use a power reduction? I suspect not since is an unknown function- not an explicit solution that allows this method..........

Thanks for reading!

(I realise the post is rather long winded, but I didnt want to miss info that the proof relied upon in previous sub part)