Problem Statement
The temperature,
, of a cooling fin of a finite length,
, satisfies
,\quad 0\leq x\leqL,\quad T(0) = 0,\quad T(L) = T_L)
where
,
, and
are constants
a) Find the complimentary function
)
, stating your answer in terms of unknown constants,
and
b)Complete the following identity
=)
c) write down your starting point for the particular solution,
)
d) find
and the coefficients
, and
and hence show that hte solution of the above problem is
 = T_L + (T_0 - T_L)\frac{\sinh(k(L-x))}{\sinh(kL)})
My attempt a)
The first thing I do here (which may be where I'm going wrong) is that I rearrange the equation. I take
 = -k^2(T_L-T))
 -k^2T(x) = -k^2T_L)
characteristic equation:
So
 = c_1 \cosh(kx) + c_2\sinh(kx) + c_3 k^2T_L)
With this, I jump straight to
 = k^2c_1\cosh(kx) +k^2c_2\sinh(kx))
I can then verify that
 -k^2T(x) + c_3k^2T_L = -k^2T_L)
is a solution, with
. But this helps me very little when I try to solve for my initial values and helps me less when I try to complete part (d).
Grinding through a bit further I can solve for
with initial values with
 = T_0 = c_1\cosh(k\cdot 0) + c_2\sinh(k\cdot 0) -k^2T_L)
So my new
)
 = (T_0 + k^2T_L)\cosh(kx)+c_2\sinh(kx)-k^2T_L)
But then attempting to solve for
I get stuck. I end up with
 = T_L = (T_0+k^2T_L)\cosh(kL)+c_2\sinh(kL)-k^2T_L)
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Originally Posted by TheBigH 
