Solving Diff. Eq: Lambda f(y), Lambda g(y) - Get Hypergeometric?

  • Thread starter VatanparvaR
  • Start date
In summary: Can anyone help me with this?Thanks!In summary, the conversation discusses two differential equations involving lambda and containing hypergeometric functions. The first equation is of second order and contains partial derivatives, while the second equation is a recurrent equation. The conversation also mentions the need to find the hypergeometric function with different parameters, and asks for resources to help with this task.
  • #1
VatanparvaR
25
0
\lambda f(y)= i b y \frac{\partial f(y)}{\partial y} + \frac{partial g(y)}{y} -\frac{k}{y}g
\lambda g(y)= i b y \frac{\partial g(y)}{\partial y} - \frac{partial f(y)}{y} +\frac{k}{y}f

I tried to get a hypergeometric eq. from these two but couldn't.
Any hints to solve?
Helps would be appreciated!
 
Physics news on Phys.org
  • #2
I somehow got this second oder diff.eq.

[tex]
(1-m^2y^2)f_{yy}-2my(i\lambda+m)f_{y}+(\lambda^2-im\lambda-\frac{k(k+1)}{y^2})f=0
[\tex]

where

[tex]f_{yy}[\tex] is [tex]\frac{\partial^2}{\partial y^2}[\tex]

Any ideas to solve this one?

p.s. Latex is not working here or am I typing wrong?
 
Last edited:
  • #3
VatanparvaR said:
[tex]\lambda f(y)= i b y \frac{\partial f(y)}{\partial y} + \frac{\partial g(y)}{y} -\frac{k}{y}g[/tex]
[tex]\lambda g(y)= i b y \frac{\partial g(y)}{\partial y} - \frac{\partial f(y)}{y} +\frac{k}{y}f[/tex]

[tex]
(1-m^2y^2)f_{yy}-2my(i\lambda+m)f_{y}+(\lambda^2-im\lambda-\frac{k(k+1)}{y^2})f=0
[/tex]

where

[tex]f_{yy} \ \mbox{is} \frac{\partial^2}{\partial y^2}[/tex]

You're using the wrong slash. The closing tag should use this "/" instead.
 
  • #4
wups, thanks very much.

and another thing, I wrote wrong the above 2 eq.s, I put + instead of minus here

[tex]\lambda g(y)= i m y \frac{\partial g(y)}{\partial y} - \frac{\partial f(y)}{y} -\frac{k}{y}f[/tex]

so it should be:

[tex]\lambda f(y)= i m y \frac{\partial f(y)}{\partial y} + \frac{\partial g(y)}{y} -\frac{k}{y}g[/tex]

[tex]\lambda g(y)= i m y \frac{\partial g(y)}{\partial y} - \frac{\partial f(y)}{y} -\frac{k}{y}f[/tex]



and then we get the above second oder diff.eq.:
[tex](1-m^2y^2)f_{yy}-2my(i\lambda+m)f_{y}+(\lambda^2-im\lambda-\frac{k(k+1)}{y^2})f=0[/tex]
 
Last edited:
  • #5
so any ideas?
 
  • #6
where
[tex]m, \lambda, k[/tex] are constants.

I am trying to put these two:
[tex]
f_1=\sum_{n=0}^{\infty}p_ny^{2n}, \ \ \ \ \ \ f_2=\sum_{n=0}^{\infty}a_ny^{2n+1}
[/tex]
and check if it is odd or even. At the end I am getting a recurrent eq.


any other ideas?
 
  • #7
hmm, it gives zero solution.
coefficients are zero in this case :(
 
  • #9
Ok, I got the solution.


Now I need one thing. From Abramowitz's book I got this one


[tex]
F(a, a+\frac{1}{2}, \frac{3}{2}, z^2)=\frac{1}{2}z^{-1}(1-2a)^{-1}[(1+z)^{1-2a}-(1-z)^{1-2a}]
[/tex]

Now I need to find

[tex]
F(a, a+\frac{1}{2}, \frac{5}{2}, z^2)
[/tex]


[tex]
F(a, a+\frac{1}{2}, \frac{7}{2}, z^2)
[/tex]


and, it would be great if I find

[tex]
F(a, a+\frac{1}{2}, n+ \frac{1}{2}, z^2)
[/tex]


are there any books, handbooks, or websites that I could find this guy?
 
  • #10
Hallooo?

Anybody is viewing this thread at all?
 
  • #11
VatanparvaR said:
wups, thanks very much.

[tex]\lambda f(y)= i m y \frac{\partial f(y)}{\partial y} + \frac{\partial g(y)}{y} -\frac{k}{y}g[/tex]

[tex]\lambda g(y)= i m y \frac{\partial g(y)}{\partial y} - \frac{\partial f(y)}{y} -\frac{k}{y}f[/tex]

You're missing two partial symbols. Are they supposed to be:

[tex]\lambda f(y)= i m y \frac{\partial f(y)}{\partial y} + \frac{\partial g(y)}{\partial y} -\frac{k}{y}g[/tex]

[tex]\lambda g(y)= i m y \frac{\partial g(y)}{\partial y} - \frac{\partial f(y)}{\partial y} -\frac{k}{y}f[/tex]

?

Also, if f and g only depend on y, then why the partials?
 
  • #12
Yeah you are right, there should be two partial symbols.

No problem with partial. As I stated above, I got the solution for this diff. eq.

[tex](1-m^2y^2)f_{yy}-2my(i\lambda+m)f_{y}+(\lambda^2-im\lambda-\frac{k(k+1)}{y^2})f=0[/tex]

from here
http://eqworld.ipmnet.ru/en/solutions/ode/ode0226.pdf

The solution, as you see, is a Hypergeometric function.

Now I need some properties of the hypergeometric function. I posted it above:

-----------
From Abramowitz's book I got this one


[tex]F(a, a+\frac{1}{2}, \frac{3}{2}, z^2)=\frac{1}{2}z^{-1}(1-2a)^{-1}[(1+z)^{1-2a}-(1-z)^{1-2a}][/tex]



Now I need to find

[tex]F(a, a+\frac{1}{2}, \frac{5}{2}, z^2)[/tex]

and

[tex]F(a, a+\frac{1}{2}, \frac{7}{2}, z^2)[/tex]


and, it would be great if I find

[tex]F(a, a+\frac{1}{2}, n+ \frac{1}{2}, z^2)[/tex]

are there any books, handbooks, or websites that I could find this guy?


Plz, help!
 
  • #13
I guess, I need to take a derivative:

[tex]
\frac{d}{dz}F(a, b, c, z^2)=\frac{ab}{2z\ c} F(a+1, b+1, c+1, z^2)
[/tex]
 

1. What is the basic concept behind solving differential equations?

The basic concept behind solving differential equations is to find a function that satisfies the given equation. This function is called the solution, and it should satisfy the equation for all values of the independent variable.

2. What is Lambda f(y) and Lambda g(y) in the context of solving differential equations?

Lambda f(y) and Lambda g(y) are both functions of the dependent variable y. They represent the coefficients of the first and second derivatives, respectively, in a differential equation. These coefficients can be constants or functions of y.

3. How is the hypergeometric function used in solving differential equations?

The hypergeometric function is a special function that is often used in solving differential equations. It is a solution to a special type of differential equation known as a hypergeometric differential equation. This function has many applications in physics, engineering, and other fields.

4. What is the process for solving a differential equation with Lambda f(y) and Lambda g(y) as coefficients?

The process for solving a differential equation with Lambda f(y) and Lambda g(y) as coefficients involves finding the general solution to the equation, which is a function that satisfies the equation for all values of the independent variable. This can be done by using various techniques such as separation of variables, substitution, or using the hypergeometric function.

5. Are there any real-world applications of solving differential equations with Lambda f(y) and Lambda g(y) as coefficients?

Yes, there are many real-world applications of solving differential equations with Lambda f(y) and Lambda g(y) as coefficients. Some examples include modeling the spread of diseases, predicting population growth, analyzing chemical reactions, and understanding the behavior of electrical circuits. These types of equations are also important in many areas of physics, engineering, and economics.

Similar threads

  • Differential Equations
Replies
1
Views
706
  • Differential Equations
Replies
3
Views
2K
  • Differential Equations
Replies
4
Views
565
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
5
Views
2K
  • Differential Equations
Replies
3
Views
1K
  • Differential Equations
Replies
7
Views
2K
Replies
3
Views
2K
  • Differential Equations
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
570
Back
Top