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Solve the following eigenvalue problem?

Posted 3 months ago
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I have two ODE and I want to find the general solution, but it contains eigenvalues. I try to solve it by using the DSolve, but could not get the solution. Please, anyone can help me to sort out the problem? It would be appreciated.

Eq1 = F'[X] + \[Lambda]^2*F[X] == 0
Eq2 = G''[Y] + (\[Epsilon]*(Y^2 Subscript[A, 1] + Y*Subscript[A, 2] + 
       Subscript[A, 3]))/Subscript[A, 4]*\[Lambda]^2*G[Y] == 0
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Suggestions.

(1) Place short code questions in the question rather than an attached notebook.

(2) Avoid upper case for function and variable names (less likely to get confused for built-in symbols).

(3) Keep the code as simple as possible-- avoid Subscript when it is not needed for example.

(4) Explain what results you got and why they are not doing what you want.

Here are the examples.

DSolve[{f'[x] + \[Lambda]^2*f[x] == 0 }, {f[x]}, x]

(* Out[2]= {{f[x] -> E^(-x \[Lambda]^2) C[1]}} *)

DSolve[{g''[y] + (\[Epsilon]*(y^2 a1 + y*a2 + a3))/
     a4*\[Lambda]^2*g[y] == 0 }, {g[y]}, y]

(* Out[3]= {{g[y] -> 
   C[1] ParabolicCylinderD[(-4 a1^(3/2) Sqrt[a4] + 
       I a2^2 Sqrt[\[Epsilon]] \[Lambda] - 
       4 I a1 a3 Sqrt[\[Epsilon]] \[Lambda])/(
      8 a1^(3/2) Sqrt[
       a4]), ((-1)^(1/4) a2 \[Epsilon]^(1/4) Sqrt[\[Lambda]])/(
       Sqrt[2] a1^(3/4) a4^(1/4)) + ((-1)^(1/4) Sqrt[2] a1^(1/4)
         y \[Epsilon]^(1/4) Sqrt[\[Lambda]])/a4^(1/4)] + 
    C[2] ParabolicCylinderD[(-4 a1^(3/2) Sqrt[a4] - 
       I a2^2 Sqrt[\[Epsilon]] \[Lambda] + 
       4 I a1 a3 Sqrt[\[Epsilon]] \[Lambda])/(8 a1^(3/2) Sqrt[a4]), 
      I (((-1)^(1/4) a2 \[Epsilon]^(1/4) Sqrt[\[Lambda]])/(
         Sqrt[2] a1^(3/4) a4^(1/4)) + ((-1)^(1/4) Sqrt[2] a1^(1/4)
           y \[Epsilon]^(1/4) Sqrt[\[Lambda]])/a4^(1/4))]}} *)

It is not clear from the post what were the desired results.

Thanks for your help. I have noted all your points. Actually, these equations (Eq.1 and 2) they came from PDE. I used separation of variables method and then equate with the Lambda so it becomes eigenvalue problem. The first equation can be solved but the second one is difficult to solve it is a Sturm-Liouville problem. I want to find the eigenvalue (Lambda) and eigenfunction. So how can I solve it? How can I solve the general solution of the equation?

PDE:

D[\[Theta][x, y], {x, 1}] == 
 Subscript[A, 
  4]/(\[Epsilon]*(Y^2 Subscript[A, 1] + Y*Subscript[A, 2] + Subscript[
     A, 3])) D[\[Theta][x, y], {y, 2}]

where A1, A2, A3,\Epsilon are constants.

Initial Condition

 \[Theta] (0, y) == 0

Boundary Condition

\[Theta]' (x, 1) == 0

Please make sure you know the rules: https://wolfr.am/READ-1ST

The rules explain how to format your code properly. If you do not format code, it may become corrupted and useless to other members. Please EDIT your post and make sure code blocks start on a new paragraph and look framed and colored like this.

int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]

enter image description here

I've not very versed in this. But you said ODE then changed your question to PDE.

However if you are using ODE "First-Order Systems" (using Matrix Algebra) in attempt to solve PDE with eigenvalues and eigenvectors, I think it would not lead to a general solution because PDE have more degrees of freedom (unless or until they are removed by trace equations and or initial conditions).

Is there some ODE or PDE book your getting this problem from the Community can refer to?

No, I have PDE and then I converted into ODE's. Actually what I am looking for, the solution of PDE. when I tried to solve my PDE it does not give me an answer so I converted into ODE's and then try to solve. I got the solution of ODE's but need to solve for eigenvalue and eigenfunction. So I stuck here.

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