Eigenvalues of the operator x d/dx

In summary: I did not mention anything about physical principles. I mentioned that the equation is a differential equation.
  • #1
eit32
21
0
a) Consider the operator x d/dx(where 1st d/dx acts on the function, then x acts on the resulting function by simply multiplying by x )acting on the set of functions of a real variable x for x>0. What are the eigenvalues and the corresponding eigenfunctions of this operator?

b) What about d/dx x (where 1st x acts on the function, then d/dx acts upon the resulting function)? What are the eigenvaules and corresponding eigenfunctions for this?
 
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  • #2
Write down the eigenvalue equation. This will be a differential equation, which you will need to solve..
 
  • #3
i don't think i quite understand what you are saying
 
  • #4
What does it mean for a function f to be an eigenvalue of x(d/dx)? What equation must it satisfy?
 
  • #5
if L is an opertaor than
L[f(x)]=cf(x); where f(x) is an eigenfunction and c is an eigenvalue
 
  • #6
Exactly. What equation do you get when you plug in the L you are given.
Do you have any idea how to solve this?
 
  • #7
x d/dx[f(x)]=cf(x)...not really
 
  • #8
OK, hint: try [tex]f(x) = x^k[/tex] for some k
 
  • #9
eit32 said:
x d/dx[f(x)]=cf(x)...not really

Why "not really"? This is correct!

In other words,

[tex] x y' = c y [/tex]

where a prime indicates a derivative with respect to x and c is some constant. This is a simple differential equation, right? Can you solve it?
 
Last edited:
  • #10
kdv said:
Why "not really"? This is correct!



[tex] x y' = c x [/tex]

I think you meant

[tex]
x y' = c y
[/tex]
 
  • #11
kamerling said:
I think you meant

[tex]
x y' = c y
[/tex]

Yes indeed. Thanks for pointing out the typo. I will correct this in my post so that it does not confuse the OP.

Thanks!
 
  • #12
yeah, i took D.E. like 3 years ago and I'm a bit hazy on the details...it looks familar but I'm not really sure where to start it
 
  • #13
eit32 said:
yeah, i took D.E. like 3 years ago and I'm a bit hazy on the details...it looks familar but I'm not really sure where to start it

It's a separable DE...

[tex] x \frac{dy}{dx} = c y \rightarrow \frac{dy}{y} = c \frac{dx}{x} [/tex]

You only need to integrate both sides
 
  • #14
Considering the quantity and type of posts you've put up at this site, eit32, you may want to dust off your DEq and linear algebra books, because most of your questions deal more with the mechanics of math operations rather than physical principles themselves.
 

1. What are eigenvalues of the operator x d/dx?

Eigenvalues of the operator x d/dx are the values of x that satisfy the equation x d/dx f(x) = λf(x), where λ is a constant and f(x) is a differentiable function.

2. How do I calculate the eigenvalues of the operator x d/dx?

The eigenvalues of the operator x d/dx can be calculated by solving the eigenvalue equation x d/dx f(x) = λf(x). This can be done by using techniques such as separation of variables or solving a differential equation.

3. What is the significance of eigenvalues of the operator x d/dx in mathematics?

Eigenvalues of the operator x d/dx are important in mathematics as they help us understand the behavior of linear operators and differential equations. They also play a crucial role in fields such as quantum mechanics and signal processing.

4. Can eigenvalues of the operator x d/dx be complex numbers?

Yes, eigenvalues of the operator x d/dx can be complex numbers. This is because the operator x d/dx is a linear operator and can have complex eigenvalues just like any other linear operator.

5. How can eigenvalues of the operator x d/dx be used in applications?

Eigenvalues of the operator x d/dx have various applications in fields such as physics, engineering, and data analysis. They are used in solving differential equations, analyzing the stability of systems, and extracting useful information from data.

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