How i find eigen function and eigen value from equation or either

[david sh]

This my data:
1.e^λx
2.e^iβx
3.sinαx
4.sin^2αx
5.cosαx
x^2

How i find eigen value and eigen function from equation above?

 

[vela]

You haven't given an equation. (Note the lack of an equal sign in your post.) Please type the problem exactly as given.

 

[david sh]

You haven't given an equation. (Note the lack of an equal sign in your post.) Please type the problem exactly as given.

number 1-6 is my equation,...

 

[vela]

None of those is an equation. To have an equation, you must have something equal to something else. You don't have that.

 

[paranormal]

To find the eigenvalues and eigenfunctions from the given equations, we first need to understand what they represent. Eigenvalues and eigenfunctions are important concepts in linear algebra and represent the values and functions that satisfy a specific equation, known as the eigenvalue equation.

In this case, the given equations do not explicitly represent an eigenvalue equation. However, we can still find the eigenvalues and eigenfunctions by manipulating the equations to fit the form of an eigenvalue equation.

For example, the first equation e^λx can be rewritten as λe^λx = 1, which is in the form of an eigenvalue equation (Ax = λx). The eigenvalues in this case would be λ = 1, and the corresponding eigenfunction would be e^x.

Similarly, the second equation e^iβx can be rewritten as iβe^iβx = 1, which also fits the form of an eigenvalue equation. The eigenvalues would be β = 1, and the corresponding eigenfunction would be e^ix = cosx + isinx.

The third equation sinαx can be rewritten as -α^2sinαx = 1, which is again in the form of an eigenvalue equation. The eigenvalues in this case would be α = ±√(1/α^2), and the corresponding eigenfunctions would be sin(√(1/α^2)x).

The fourth equation sin^2αx can be rewritten as (1-α^2)sin^2αx = 1, which is also in the form of an eigenvalue equation. The eigenvalues would be α = ±√(1/(1-α^2)), and the corresponding eigenfunctions would be sin(√(1/(1-α^2))x).

The fifth equation cosαx can be rewritten as -α^2cosαx = 1, which is once again in the form of an eigenvalue equation. The eigenvalues in this case would be α = ±√(-1/α^2), and the corresponding eigenfunctions would be cos(√(-1/α^2)x).

Finally, the equation x^2 can be rewritten as x^2 = λx, which is also in the form of an eigenvalue equation. The eigenvalues in this case would be λ = 0, and the corresponding eigenfunctions would be x.

In