Applying Initial Conditions

I need to find all the separated solns of

[tex] x^2 \frac{\partial^2 u}{\partial x^2} + x\frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial y^2} = 0 [/tex]

in the strip [tex]{(x,y) : 0 < y < a, -\infty < x < \infty } [/tex]
the separated solns must also satisfy u = 0 on both the edges, that is, on y=0 and y=a for all values of x.

Iv got the general solutions to be..

[tex] X(x) = Dlnx + C , (k = 0) [/tex]
[tex]X(x) = Dx^{n} + Cx^{-n} , (k \neq 0) [/tex]

and

[tex]Y(y) = A\cos{ky} + B\sin{ky} , (k \neq 0)[/tex]
[tex]Y(y) = Ay + B , (k = 0)[/tex]

where k is just the constant iv let the two bits equal when I separated the variables. (well -k^2 actually).

I just need help interpreting the conditions to sort out the constants..I think!
 

Galileo

Science Advisor
Homework Helper
1,989
6
I haven't checked your answer, but if it is correct then, since [itex]u(x,y)=X(x)Y(y)[/itex], the boundary conditions say:

[tex]u(x,0)=X(x)Y(0)=0[/tex]
and
[tex]u(x,a)=X(x)Y(a)=0[/tex]

So [itex]Y(0)=Y(a)=0[/itex]

For example: if k=0, then applying the boundary condition at y=0 gives:
[tex]Y(0)=B=0[/tex]
 

Related Threads for: Applying Initial Conditions

  • Posted
Replies
13
Views
4K
Replies
2
Views
4K
Replies
10
Views
855
  • Posted
Replies
1
Views
760
  • Posted
Replies
2
Views
3K
Replies
1
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top