I need to solve the antiderivative I = int(ArcTg[Ln(x) + C])?

Question

This indefinite integral comes from the differential equation
x*y'' = y' + [y']^3. The problem is that my professor told me a way of solving it, but there is a problem with his method though.

OK, now: The first thing is the reduction of order in the differential equation.

If you make the following changes: y' = p AND y'' = (dp/dx)p, what you obtain is x*p(dp/dx) = p +(p^3). If you separate variables, solve for p, then revert the changes and solve for y, you derive my question. BUT, according to the book of Simmons, the answer is C1^2 = x^2 + [y - C2^2]^2, but I do not believe that solving the antiderivative I'll manage to get that!!!

The funny thing is that you DO solve the equation and get the answer if you make the changes: y' = p AND y'' = dp/dx... hahahaha, oh damn!, sorry I have already figured out my mistake, I applied another method. But I would really appreciate if someone could solve the initial question please (the antiderivative).

Thanx!!

Answer 1

Hey.

I was reading your answer and I was about to ask you why you had had this variable change the way that you mentioned in the first part of your question. Yesterday night I was thinking why. Now I read the rest of it and I noticed that you had made a mistake. OK

So lets now talk about the integral part. I really dont think that there is a function that is the antiderivative of this function.

So, I suggest you to use Taylos series and integrate them if you really need to calculate it sometime in the future.

You can write a computer program to calculate the definited integral too.

Or you can take derivatives and then integrate, this functions in some exercises.

Ana

Answer 2

Its too late, but I will check your question tomorrow

Ana