Solving 2nd Order ODE: Even Function Solution

In summary, the conversation is discussing a differential equation with the form of a second derivative of y plus a constant term multiplied by y equals zero. The person is looking for an even function solution and mentions encountering this equation while studying surface waves in a self-gravitating incompressible fluid with stratification. They suggest using Mathematica or Wolfram Alpha to find a solution and mention that the resulting equation can be written in terms of the associated Legendre's equation.
  • #1
omyojj
37
0
Could you please help me or give me any hint to solve this ODE..

[tex]\frac{d^2y}{d x^2} + ( 2\rm{sech}^2 x - a^2 ) y = 0[/tex]

where a is a constant.

I want only even function solution. (y(x) = y(-x))

BTW, this is a homework problem. I encountered this equation while considering surface waves in a self-gravitating incompressible fluid with stratification.
 
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  • #2
You probably need to post this then in the homework forum where you are required to show some effort. Look, this is what I would do: I'd study it for a while, see if I can do something about it, I couldn't initially, then just resort in desperation to Mathematica. Now, if you don't have it, then you either need to find a machine running it or just plug it into Wolfram Alpha. Mathematica gives a solution and that solution is in terms of a particular well-known DE. At that point you can either just use it if you're an engineer or something, or if you like math, you might try and figure how to convert your DE into that particular DE for which the solution is given in terms of and therefore "figure" how to solve it. That is, show how your equation can be written as:

[tex](1-u^2)y''-2uy'+\left(2-\frac{a^2}{1-u^2}\right)y=0[/tex]

for [itex]u=f(x)[/itex] for appropriate [itex]f(x)[/itex].
 
Last edited:
  • #3
Oops, I committed an error...was going to say 'this is NOT a homework problem'..

[tex]x = \mathrm{tanh}u[/tex]

and the resulting equation is the associated Legendre's equation.
Thank you anyway, I should've examined the equation with more patience..
 

Related to Solving 2nd Order ODE: Even Function Solution

1. What is a 2nd Order ODE?

A 2nd Order ODE (Ordinary Differential Equation) is a mathematical equation that involves a function and its first and second derivatives. It is typically written in the form of y'' + P(x)y' + Q(x)y = f(x), where y is the function, P and Q are coefficients, and f(x) is a known function. Solving a 2nd Order ODE involves finding the function y that satisfies the equation.

2. What is an Even Function?

An even function is a mathematical function that satisfies the property f(-x) = f(x) for all values of x. In other words, an even function is symmetric about the y-axis, so its graph is unchanged when reflected across the y-axis. Examples of even functions include cos(x) and x2.

3. How do I solve a 2nd Order ODE with an even function solution?

To solve a 2nd Order ODE with an even function solution, you can use the method of undetermined coefficients. This involves assuming a general form for the solution, plugging it into the ODE, and then solving for the coefficients. For an even function solution, the general form would be y = Ax2 + Bx. Once you have the coefficients, you can substitute them back into the general form to get the specific solution.

4. Are there any special cases when solving a 2nd Order ODE with an even function solution?

Yes, there are two special cases to be aware of when solving a 2nd Order ODE with an even function solution. The first is when the ODE includes a term involving y', in which case the general form of the solution would be y = Ax2 + Bx + C. The second is when the ODE has a term involving y'', in which case the general form would be y = Ax2 + Bx + Cx3.

5. How do I check if my solution to a 2nd Order ODE with an even function solution is correct?

To check if your solution to a 2nd Order ODE with an even function solution is correct, you can plug it back into the ODE and see if it satisfies the equation. Additionally, you can graph the solution and see if it matches the behavior of the ODE. Another option is to use a computer program or calculator to numerically solve the ODE and compare the results to your solution.

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