separating variables! help!

ntfabolous · #1 $Old$ October 15th, 2009, 09:55 PM

Sorry, but this is an upcoming exam question for me. I need help!(and I only got a short time to find out)

Find solutions of the following equation by separating variables:

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$

Thanks to whoever that can help!

HallsofIvy · #2 $Old$ October 16th, 2009, 05:56 AM

Quote:

Originally Posted by ntfabolous $View Post$

Sorry, but this is an upcoming exam question for me. I need help!(and I only got a short time to find out)

Find solutions of the following equation by separating variables:

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$

Thanks to whoever that can help!

Do you know what "separating variables" means? If you have a test coming up, this is hardly the time to start learning the subject!

It means assuming a solution of the form u(x,y)= A(x)B(y) where A and B are functions of x alone and y alone, respectively. Then $\frac{\partial u}{\partial x}= \frac{dA}{dx}B$ and $\frac{\partial u}{\partial y}= A\frac{dB}{dy}$ .

So that equation becomes $\frac{dA}{dx}B+ A\frac{dB}{dy}= 0$ . That can be written as $\frac{dA}{dx}B= -A\frac{dB}{dy}$ and, dividing both sides by AB, $\frac{1}{A}\frac{dA}{dx}= -\frac{1}{B}\frac{dB}{dy}$ .

Now, the left side is a function of x only and the right side is a function of y only (we have "separated" the variables) so the only way that can be equal for all x and y is if they are both equal to the same constant.

That is, we can separate into two equations: $\frac{1}{A}\frac{dA}{dx}= \lambda$ or $\frac{dA}{dx}= \lambda A$ and $-\frac{1}{B}\frac{dB}{dy}= \lambda$ or $\frac{dB}{dy}= -\lambda B$ .

There is no way to determine what " $\lambda$ " is from the equation alone. Depending upon the additional (boundary or intial value) conditions, the solution might be the product of A and B for a specific $\lambda$ or a sum of such products for many (possibly infinitely many) values of $\lambda$ .

Danny · #3 $Old$ October 16th, 2009, 07:29 AM

Quote:

Originally Posted by ntfabolous $View Post$

Sorry, but this is an upcoming exam question for me. I need help!(and I only got a short time to find out)

Find solutions of the following equation by separating variables:

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$

Thanks to whoever that can help!

I might also add the one could seek solutions of the form

$u = A(x) + B(y)$

which is also a separation of variables. BTW - the general solution of the above is $u = f(x-y)$ for any $f$ .