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