What is the Integral Solution for a Calculus Problem with Two Variables?

[eas123]

Homework Statement

See attached.

Homework Equations

The Attempt at a Solution

I integrated the equation with respect to x to obtain
∫\frac{d}{dx}(xe^{-x}\frac{df}{dx})dx+∫ne^{-x}fdx= constant
The first term on the left hand side goes to zero as x, df/dx are bounded at 0, infinity. This leaves the expression ∫ne^{-x}fdx= constant which is not the one given.

 

[tiny-tim]

hi eas123!

what happened to m ? ​

 

[eas123]

Hi. :-)

What do you mean? I don't know how to derive the expression.

 

[tiny-tim]

your expected answer, $\int_0^{\infty} e^{-x}f_n(x)f_m(x) dx = 0$, has an "m" in it

i don't see an "m" in your actual work

 

[eas123]

So where have I gone wrong?

 

[tiny-tim]

So where have I gone wrong?

i'm completely confused

you seem to be solving a different problem

start with $e^{-x}f_n(x)f_m(x) dx = 0$, and integrate it ​

 

[SammyS]

Homework Statement

See attached.

Homework Equations

The Attempt at a Solution

I integrated the equation with respect to x to obtain
∫\frac{d}{dx}(xe^{-x}\frac{df}{dx})dx+∫ne^{-x}fdx= constant
The first term on the left hand side goes to zero as x, df/dx are bounded at 0, infinity. This leaves the expression ∫ne^{-x}fdx= constant which is not the one given.

The following is an image of the attachment. Please notice that the result you are to prove contains both n and m . That's what tiny-tim is telling you.