Question

Seperate the Partial fraction equation

Answer 1

\[u _{x _{x}}-c^2k^2u _{y _{y}} \]

Answer 2

It is a 2D heat equation

Answer 3

isn't that partial differential equation??

Answer 4

yes it is

Answer 5

@TuringTest @Zarkon @jim_thompson5910

Answer 6

I don't see an equals sign
you say this is the 2d Heat equation?

Answer 7

here is a good example of solving the heat equation by separation of variables if that is what you wanted:
http://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx

Answer 8

it is equal to zero

Answer 9

@TuringTest

Answer 10

then you have the heat equation as\[u_{xx}=c^2k^2u_{yy}\]which confuses the heck out of me; where is time and why are they both second derivatives?
not how I know the heat equation...

Answer 11

\[u_t=ku_{xx}\]is how I know it

Answer 12

I suppose k=c^2k^2 can reconcile part of it, but still your formula makes no sense to me
what are x and y ?

Answer 13

wasiqss.. that's not a heat equation... its a wave equation, a hyperbolic one.. may I request you to rewrite it :))

Answer 14

lol @anonymoustwo44 the question is just written as , what i wrote, and heading that solve the Partial Differential equation

Answer 15

do you mean:

Answer 16

tomorow is my exam, and this quetsion is expected plz can you solve this for me

Answer 17

u_xx-c^2k^2u_yy=0?

Answer 18

yup

Answer 19

ah so its a wave equation...

Answer 20

ok. fine then

Answer 21

:)) ok what are your Boundary value conditions or initial conditions ?

Answer 22

its just written that , seperate the variables and find solution u(x,y)

Answer 23

ok ok so you are to use seperation of variables right?

Answer 24

yup

Answer 25

ok so we assume the solution is in the form u(x,y)=w(x)q(y)

Answer 26

yup

Answer 27

and so we have
u_xx-c^2k^2u_yy=0
q(w_xx)-c^2k^2w(q_yy)=0
q(w_xx)=c^2k^2w(q_yy)
(w_xx)/w=c^2k^2(q_yy)/q=s
now we have
(w_xx)/w=s c^2k^2(q_yy)/q=s
w_xx=ws q_yy=qs/c^2k^2
w_xx-ws=0 q_yy-qs/c^2k^2=0

and now you have two ordinary second order differential equation

Answer 28

should I continue?

Answer 29

yes plz im too sick to do, high fever

Answer 30

@anonymoustwo44 use this to build latex equation fast and easy if you have prob
http://www.codecogs.com/latex/eqneditor.php

Answer 31

well, I'll be assuming first that c^2k^2>0. now, find solution for 3 cases. for first case, we have s<0: so you find general solution for q and w in this case then multiply them both to find u(x,y) when s<0. then second c=0: again find general solution for w(x) and q(y) then multiply again them to get u(x,t) for x=0. then the last case c>0: find general solution for w(x) and q(y) then multiply to get u(x,y) for s>0.
Now, add all the solutions you obtain to get the general solution for u(x,y). now most of the times, for each cases, you'll be computing for eigensolutions and their corresponding eigenvalues (there are infinite number of them in most cases)) for each case. Then we list all of the possible eigensolutions, then we add them all up. And then we get an infinite series which looks a lot like the form of the fourier series. Now using the initial conditions or boundary value conditions, we find the constants in the series. you may check this site: tutorial.math.lamar.edu . The solution most of the times are REALLY MESSY :)) and it may involve a lot of work.

Answer 32

its really hard to type all that equation to this site. its really REALLY long. But if you will insist, I will give the solution tomorrow