ODE Approaching the expicit solution

[freestar]

Hi there,
This ode has me really stumped. Since it is non linear, I don't know which method to use for this:

$$xy' + 2y = \frac{sec^2(y)}{x}$$

Thank you :)

 

[coelho]

There are two hints that may be useful (or not).

$\frac{d(tan(y))}{dy}=?$

$\frac{d(x^2 y)}{dx}=?$

 

[fluidistic]

There are two hints that may be useful (or not).

$\frac{d(tan(y))}{dy}=?$

$\frac{d(x^2 y)}{dx}=?$

I'm not the OP but I'm interested to know how this can help. I calculated both of the derivatives and I notice something (not sure I can tell) but is that supposed to make the DE separable? Because it doesn't, to me. :/

 

[coelho]

This small hints aren't supposed to make the DE separable. They are just things i noticed when looking to the DE, things that beginners usually don't notice, and that may put him (or her) one step closer to the solution, as we aren't supposed to give straightaway answers or hints that make they work too easy.

 

[JJacquelin]

Hello !

I don't think that the ODE can be analytically solved in using only standard functions.
So, I strongly suggest to use numerical methods instead of searching an explicit solution.
Nevertheless, approximate analytic solutions can be obtained, depending on the range where they are derived. An example is given in attachment.

 

[fluidistic]

I'm also totally stuck in solving this DE in analytical form. Maybe coelho could give 1 more hint. Also since this question isn't in the homework section, it's supposedly not a coursework question and giving huge hints shouldn't be a problem (IMO).