Need a solution for the following problem

  • Thread starter Showstopper
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In summary, the conversation is about finding the differential equation for F(x) given that F(0) is equal to F0, which can be any constant but must be explicitly determined. The differential equation is either \frac{dF}{d\omega}= h(\omega)F or \frac{dF}{d\omega}+ h(\omega)F= 0 depending on the sign of h. The value of F0 is determined to be \sqrt{\pi}. The problem does not ask to solve the differential equation, but rather to find an equation of that form for F(x). The function F(x) is defined as F(\omega)= \int_{-\infty}^{\infty}
  • #1
Showstopper
7
0
The question is attached. Thanks a lot
 

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  • #2
Expressed as an integral, what is F(0) equal to?
 
  • #3
F(0) = Fo where Fo can be any constant, but we have to specify it.
 
  • #4
not specify but determine actually my bad
 
  • #5
I SAID:
Expressed as an integral, WHAT IS F(0)?
 
  • #6
see attached
 

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  • #7
Indeed!
And what is the exact value of that integral?
 
  • #8
- e of positive infinity??
 
  • #9
Okay, so you are unfamiliar with the famous result:
[tex]\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}[/tex]
 
  • #10
yea, never seen this before
 
  • #11
Okay, so now you know the value of F(0)! :smile:
 
  • #12
So what is the answer?
Care to walk me through?
 
  • #13
arildno said:
Okay, so now you know the value of F(0)! :smile:

So what do we do with this?
Sorry, I am not understanding.
 
  • #14
rad0786 said:
So what do we do with this?
Sorry, I am not understanding.

The problem SAID "Write a differential equation for F(x):
[tex]\frac{dF}{d\omega}+ h(\omega)F[/tex]
with F(0)= F0 where F0 is the constant you have to determine explicitly."

Okay, you now know that F0= [itex]\sqrt{\pi}[/itex].

By the way, I notice that the "differential equation" you give (I have copied it exactly above) is not an equation! There is no equal sign. I imagine it is actually either
[tex]\frac{dF}{d\omega}= h(\omega)F[/tex]
or
[tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex]

They differ, of course, only in the sign of h.
 
  • #15
HallsofIvy said:
The problem SAID "Write a differential equation for F(x):
[tex]\frac{dF}{d\omega}+ h(\omega)F[/tex]
with F(0)= F0 where F0 is the constant you have to determine explicitly."

Okay, you now know that F0= [itex]\sqrt{\pi}[/itex].

By the way, I notice that the "differential equation" you give (I have copied it exactly above) is not an equation! There is no equal sign. I imagine it is actually either
[tex]\frac{dF}{d\omega}= h(\omega)F[/tex]
or
[tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex]

They differ, of course, only in the sign of h.


Oh so we just have to solve the ODE [tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex] with initial condition F0= [itex]\sqrt{\pi}[/itex]...

is it that straight forward? Am I not understanding something?
 
  • #16
but how do you solve a question like this, with 3 variables and a complex number? even if we use the method of solving linear differential equations
 
  • #17
rad0786 said:
Oh so we just have to solve the ODE [tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex] with initial condition F0= [itex]\sqrt{\pi}[/itex]...

is it that straight forward? Am I not understanding something?

Apparently you are having trouble reading the problem- which I just quoted.

The problem does NOT ask you to solve any differential equation. It asks you to FIND an equation of that form (essentially find the function [itex]h(\omega)[/itex] so that F(x), as given in integral form, satisfies that equation.

What happens if you differentiate the integral defining F?
 
  • #18
I got F(w) = root( pi) when solving the diff. equation

So is this the answer to the integral?
 
  • #19
This is becoming very frustrating. Is there any point in responding if you don't read the replies?? I just said, the problem does not ask you to solve a differential equation, it asks you to find a differential equation for F! And I don't know what you mean by "the answer to this integral". Integrals don't have answers, questions do. What is the question?

You are told that F is defined by
[tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex]
What do you get if you differentiate that equation with respect to [itex]\omega[/itex]? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, [tex]e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}[/tex]
 
  • #20
HallsofIvy said:
This is becoming very frustrating. Is there any point in responding if you don't read the replies?? I just said, the problem does not ask you to solve a differential equation, it asks you to find a differential equation for F! And I don't know what you mean by "the answer to this integral". Integrals don't have answers, questions do. What is the question?

You are told that F is defined by
[tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex]
What do you get if you differentiate that equation with respect to [itex]\omega[/itex]? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, [tex]e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}[/tex]

I applogize, it's not every day I come across questions like this...this is very challenging.

I got confuzed between "solving the differential equation" and "finding the differenatial equation of F" -- I don't know the difference. :uhh:

The derivative of [tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex] with respect to [itex]\omega[/itex] is [tex]F(\omega)= \int_{-\infty}^{\infty}-xe^{-\omega x}e^{-x^2}dx[/tex]
 
Last edited:
  • #21
Stupid latex...that's not the integral...latex keeps giving the wrong one
 

1. What is the problem that needs a solution?

The problem is not specified in this question. You will need to provide more information or context in order for a solution to be determined.

2. Is this a scientific problem?

It depends on the nature of the problem. If it involves a scientific inquiry or requires a scientific approach to find a solution, then it can be considered a scientific problem.

3. Can you provide more details about the problem?

Yes, in order for a solution to be determined, it is important to have a clear understanding of the problem and its specific details. This will help in identifying potential solutions.

4. What is the desired outcome or goal of finding a solution?

The desired outcome can vary depending on the problem. It could be to improve a process, solve a technical issue, or achieve a certain result. Identifying the desired outcome will help in finding the most suitable solution.

5. How can a solution be determined?

A solution can be determined through a systematic and logical approach that involves analyzing the problem, gathering information, brainstorming potential solutions, and evaluating their feasibility and effectiveness. It may also involve experimentation or testing to validate the chosen solution.

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