# Separation of Variables for ODE

1. Apr 13, 2012

### spaghetti3451

1. The problem statement, all variables and given/known data

Solve the following equation by separation of the variables:

y' tan-1x - y (1+x2)-1 = 0

2. Relevant equations

3. The attempt at a solution

I am not sure if tan-1x stands for arctan x or (tan x)-1. (This has been taken out a book.) Any help on this would be appreciated.

If arctan x, then we have to integrate 1 / [(1+x2)(arctan x)] w.r.t x. No idea how to do this.

If (tan x)-1, then we have to integrate tan x / (1+x2) w.r.t. x. No idea on how to do this either.

2. Apr 13, 2012

It's should be arctan(x). So
$$y'\arctan(x)-\frac{y}{1+x^2}=0\\ \Rightarrow \int \frac{\mathrm{d}y}{y}=\int \frac{1}{1+x^2}\frac{1}{\arctan(x)}\, \mathrm{d}x\text{.}$$
At this point it is helpful to note that
$$\frac{\mathrm{d}}{\mathrm{d}x}\arctan(x)=\frac{1}{1+x^2}\text{.}$$
So, your integral will just be a $u$ substitution. I believe you can take it from here?

3. Apr 13, 2012

### spaghetti3451

Yeah, I think I can do the rest. Sub u = arctan(x) and find the answer to be y = k arctan (x).

4. Apr 13, 2012