Suppose satisfies the equation Find a closed-form expression for
Note: Don't worry about details such as where and are defined, etc. You may assume whatever is needed!
Setting we have...
(1)
... with the condition...
(2)
Since is ...
(3)
... the condition (2) becomes...
(4)
... that can be written as...
(5)
The (5) is a first order ODE and we have to find a solution that for (1) has to be analythic... end of the first step ...
Kind regards
this is probably the most interesting problem i've ever posted in MHF and i was kind of disappointed to see that it was ignored! so, thanks for replying!
ok, your last result, (5), is not correct. it should be i'll call this (1). from it's clear that and so from (1) we have
now using Leibniz rule (n-th derivative of product of two functions) in (1) we'll get: call this (2).
let we already showed that so if, in (2), we put we'll get: we will call this (3).
finally use induction in (3) to show that (this won't be an easy induction! also we define so the formula for will be valid for )
therefore
The ODE we arrived in previous post [and that I have corrected...] can be written simply as...
(1)
... and now we have to search a solution of it of the form...
(2)
The 'brute force' approach to (1) is difficult because it is a non linear ODE, som that we swap the role of variables x and y 'transforming' the (1) in ...
(3)
The (3) is a linear ODE and its general solution is easy to find...
(4)
But for (2) must be so that is and the inverse of the function we are searching is...
(5)
... and we don't achieve any progress because (5) is not a surprise!... never mind!...
Kind regards