partial differential equations help!!non homog

[theverymooon]

Hello, I here is my heat equation problem

Ut(x,t) = Uxx(x,t) - bU(x,t)
U(0,t) = 0
U(pi, t) = 1
U(x,0) = 0

Now i need to show the equilibrium solution is Uinfinity(x) = sinh(x*sqrt(b)) / sinh (pi*sqrt(b))
I can see that this is true by plugging in the boundary conditions but im pretty sure thats not what i need to to. I am assuming i need to solve an ODE
Plugging in Uinfinity into the first problem i end up with
U(infinity)" = b*U(infinity) ie f"(x) = b*f(x)

How can this be? Am i doing something wrong? The only thing i can think of is that f(x) (ie Uinfinity) is equal to zero.
Sorry for the sloppy notation. I have only used this site once before and im not sure how to input the proper math notation. Tips on that would be appreciated as well.
Thanks!!!

 

[theverymooon]

ok so now i see that they can be equal with sinh and cosh derivatives, but how do they pull that out of nowhere? how do we know it is not sin and cos??
anyone please?

 

[DrMike]

Hello, I here is my heat equation problem

Ut(x,t) = Uxx(x,t) - bU(x,t)
U(0,t) = 0
U(pi, t) = 1
U(x,0) = 0

Now i need to show the equilibrium solution is Uinfinity(x) = sinh(x*sqrt(b)) / sinh (pi*sqrt(b))
I can see that this is true by plugging in the boundary conditions but im pretty sure thats not what i need to to. I am assuming i need to solve an ODE
Plugging in Uinfinity into the first problem i end up with
U(infinity)" = b*U(infinity) ie f"(x) = b*f(x)

How can this be? Am i doing something wrong? The only thing i can think of is that f(x) (ie Uinfinity) is equal to zero.
Sorry for the sloppy notation. I have only used this site once before and im not sure how to input the proper math notation. Tips on that would be appreciated as well.
Thanks!!!

At equilibrium, U[sub:3w4hbcj0]t[/sub:3w4hbcj0]=0. This gives U[sub:3w4hbcj0]xx[/sub:3w4hbcj0](x)-bU(x) = 0. When you solve this, and substitute your boundary conditions, you should be able to rearrange the solution into the given form.

NB - remember, when solving it, that you don't *yet* know whether b>0, b<0 or b=0.

 

[theverymooon]

Hey, thank you for replying.
I know that i have to solve Uxx - bU = 0
But i just dont understand how they are pulling the sinh and cosh out of nowhere. How do we know its not sin and cos? Maybe im missing a crucial step here, but i just dont see how to get it.
i see we can solve s^2 - b = 0
==> s = +- sqrt(b) so we don't know if b >0 b< 0 or b=0
so do we need to do cases? if so, i still don't see how that gives us the desired solution sinh(x*sqrt(b)) / sinh (pi*sqrt(b)
i guess i can see b can't be zero since we can't have zero in the denominator, but how do we decifer the rest? also sinh(pi*sqrt b) is a constant how does that work?
ive dont other problems like this where you find an arbitrary Uinfinity" and integrate but that wouldn't work would it? im thinking here that we have to find it using the eigen value technique, but ive never seen it done before like that. Sorry im kind of thinking outloud here but i just want you to see where i am coming from. i tend to type like i would talk. Thank you in advance for your help!!

 

[Deleted member 4993]

ok so now i see that they can be equal with sinh and cosh derivatives, but how do they pull that out of nowhere?

If you differentiate the "hyperbolic" forms - you'll see that those satisfy your given DE.

how do we know it is not sin and cos?? If you differentiate the "sine/cosine" forms - you'll see that those do not satisfy your given DE.

anyone please?

The solution is formed in a way similar to the solution of:

y" + w[sup:2vpej31t]2[/sup:2vpej31t]x = 0

y = Acos(wx) + Bsin(wx)