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Behaviour of implicit ODE solution as x approaches infinity

  • Thread starter bmxicle
  • Start date
  • #1
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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 [Broken]



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 [Broken]

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 [Broken]

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.
 
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