Need a solution for the following problem

[Showstopper]

The question is attached. Thanks a lot

 

[arildno]

Expressed as an integral, what is F(0) equal to?

 

[Showstopper]

F(0) = Fo where Fo can be any constant, but we have to specify it.

 

[Showstopper]

not specify but determine actually my bad

 

[arildno]

I SAID:
Expressed as an integral, WHAT IS F(0)?

 

[Showstopper]

see attached

 

[arildno]

Indeed!
And what is the exact value of that integral?

 

[Showstopper]

- e of positive infinity??

 

[arildno]

Okay, so you are unfamiliar with the famous result:
\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}

 

[Showstopper]

yea, never seen this before

 

[arildno]

Okay, so now you know the value of F(0)!

 

[rad0786]

So what is the answer?
Care to walk me through?

 

[rad0786]

Okay, so now you know the value of F(0)!

So what do we do with this?
Sorry, I am not understanding.

 

[HallsofIvy]

So what do we do with this?
Sorry, I am not understanding.

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

Okay, you now know that F0= \sqrt{\pi}.

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
\frac{dF}{d\omega}= h(\omega)F
or
\frac{dF}{d\omega}+ h(\omega)F= 0

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

 

[rad0786]

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

Okay, you now know that F0= \sqrt{\pi}.

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
\frac{dF}{d\omega}= h(\omega)F
or
\frac{dF}{d\omega}+ h(\omega)F= 0

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

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

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

 

[Showstopper]

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

 

[HallsofIvy]

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

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 h(\omega) so that F(x), as given in integral form, satisfies that equation.

What happens if you differentiate the integral defining F?

 

[rad0786]

I got F(w) = root( pi) when solving the diff. equation

So is this the answer to the integral?

 

[HallsofIvy]

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
F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx
What do you get if you differentiate that equation with respect to \omega? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}

 

[rad0786]

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
F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx
What do you get if you differentiate that equation with respect to \omega? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}

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.

The derivative of F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx with respect to \omega is F(\omega)= \int_{-\infty}^{\infty}-xe^{-\omega x}e^{-x^2}dx

 

[rad0786]

Stupid latex...that's not the integral...latex keeps giving the wrong one