Behaviour of implicit ODE solution as x approaches infinity

[bmxicle]

Homework Statement

This is the solution to an IVP, and the question asks how the function behaves as x Approaches infinity.
[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Carctan%20x%5C%2C%20%2B%5C%2C%20%20x%5Csin%20y%20%5C%2C%2B%5C%2C%20%5Cfrac%7By%5E3%7D%7B6%7D%5C%2C%20%3D%5C%2C%20%5Cfrac%7B2%5Cpi%5E3%5C%2C%2B%5C%2C%203%5Cpi%7D%7B12%7D.gif

Homework Equations

n/a

The Attempt at a Solution

Well i checked the solution on wolfram alpha and it gave me the same solution, so I'm fairly confident my solution is correct and that it wasn't supposed to be a function with an easier limit to compute.

rearranging the equation gives.


[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Csin%20y%20%5C%2C%20%2B%5C%2C%5Cfrac%7By%5E3%7D%7B6x%7D%20%5C%2C%20%3D%20%5C%2C%20%5Cfrac%7B2%5Cpi%5E3%20%2B3%5Cpi%7D%7B12x%7D%5C%2C%20-%5C%2C%20%5Cfrac%7B%5Carctan%20x%7D%7Bx%7D%5C%5C%5Clim_%7Bx%5Cto%5Cinfty%7D%20%5C%2C%20%5Cfrac%7B2%5Cpi%5E3%20%2B3%5Cpi%7D%7B12x%7D%5C%2C-%5C%2C%20%5Cfrac%7B%5Carctan%20x%7D%7Bx%7D%5C%2C%20%3D%20%5C%2C%200%20.gif

The limit of the left hand side of the equation must then be:

[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Clim_%7Bx%5Cto%5Cinfty%7D%20%5Csin%20y%5C%2C%20%2B%5C%2C%5Cfrac%7By%5E3%7D%7B6x%7D%20%5C%2C%20%3D%20%5C%2C%200%20%5C%5C%20%5Cleftrightarrow%20%5C%2C%20y%5C%2C%20%3D%5C%2C%200.gif

So since the right hand side of the solution's limit is zero, for the lefthand side of the solution's limit must also be equal to zero and this occurs iff y = 0.

I'm just not sure if this is a valid approach to computing 'limits' of implicit functions as I don't have Much experience with them yet.

 

[Mark44]

This seems reasonable to me, except that you can't conclude that y necessarily is 0. Here's a plot on wolframalpha that seems to show horizontal asymptotes at 0, pi and 2pi.
http://www.wolframalpha.com/input/?...=+Pi+(3+++2+Pi^2),+{x,+100,+200},+{y,+0,+15}]

Notice that sin(0) = 0, but also sin(pi) = 0 and sin(2pi) = 0. Since x is getting large, if y is close to any of these three values, y^3/6x --> 0 as well.