ODE Approaching the expicit solution

  • Thread starter freestar
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In summary, the conversation is about a non-linear ode that the participants are struggling to solve. They mention two hints that may be helpful in solving the ode, but it is not clear if they are able to use them to make the ode separable. The conversation also touches on the possibility of using numerical methods to solve the ode and the potential for obtaining approximate analytic solutions.
  • #1
freestar
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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 :)
 
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  • #2
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]
 
  • #3
coelho said:
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. :/
 
  • #4
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.
 
  • #5
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.
 

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  • #6
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).
 

1. What is an ODE?

An ODE, or Ordinary Differential Equation, is a mathematical equation that relates a function to its derivatives. It is used to model many real-life phenomena such as motion, growth, and chemical reactions.

2. What does it mean for an ODE to have an explicit solution?

An ODE has an explicit solution if the dependent variable can be expressed explicitly in terms of the independent variable and any parameters in the equation. This means that the solution can be written in a closed form, without the need for numerical methods to approximate it.

3. How do you approach finding the explicit solution of an ODE?

The first step is to identify the type of ODE (linear, separable, exact, etc.) and use the appropriate method to solve it. This may involve integrating both sides of the equation, using substitution, or other techniques. Once the general solution is found, it can be further simplified by applying initial or boundary conditions.

4. What are the advantages of using an explicit solution for an ODE?

Having an explicit solution allows for a deeper understanding of the behavior of the system modeled by the ODE. It also allows for easier interpretation and visualization of the solution. Furthermore, explicit solutions can be used to make predictions and analyze the effects of different parameters on the system.

5. Are there any limitations to using explicit solutions for ODEs?

Explicit solutions may not always exist for certain types of ODEs, such as nonlinear or non-separable equations. In addition, even if an explicit solution exists, it may not be possible to find it analytically and numerical methods may be necessary. Lastly, explicit solutions may not be accurate in certain situations and may require further approximations or adjustments.

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