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:

[tex]xy' + 2y = \frac{sec^2(y)}{x}[/tex]

Thank you :)

coelho

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

[itex]\frac{d(tan(y))}{dy}=?[/itex]

[itex]\frac{d(x^2 y)}{dx}=?[/itex]

fluidistic

Quote by coelho

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

[itex]\frac{d(tan(y))}{dy}=?[/itex]

[itex]\frac{d(x^2 y)}{dx}=?[/itex]

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 dont 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).

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freestar	ODE Approaching the expicit solution Tweet Hi there, This ode has me really stumped. Since it is non linear, I don't know which method to use for this: [tex]xy' + 2y = \frac{sec^2(y)}{x}[/tex] Thank you :)

coelho	There are two hints that may be useful (or not). [itex]\frac{d(tan(y))}{dy}=?[/itex] [itex]\frac{d(x^2 y)}{dx}=?[/itex]

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. Attached Thumbnails

fluidistic Recognitions: Gold Member	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).