(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Folks, I am self studying through a heat conduction problem involving a 2nd order linear homogenous differential equation which has the solution of the form

##\theta (x)=C_1\cosh mx+ C_2\sinh mx## (1)

where ##m \equiv \sqrt \frac{c}{a}= \sqrt{\frac{\beta P}{k A}} ##

The constants are dertermined via the BC's ##\theta(0)=\theta_0## and

##[\theta_x+\frac{\beta}{k} \theta]_{x=l}=0## using ##sinh x =(e^x-e^{-x})/2## etc etc.

I can determine ##C_1=\theta(0)## but I dont know how ##C_2## is determined using the hyperbolic expression....

3. The attempt at a solution

I attempted to rearrange ##\theta_x=-\frac{\beta}{k} \theta## from the BC given and equate that to the derivative of the general form of solution and then subsitute x=l in order to find C_2....ie

##\theta'(x)= m C_1 \sinh mx +C_2 \cosh mx## therefore at x=L

##\theta'(L)=m \theta_0 \sinh m L +C_2 \cosh m L=-\frac{\beta}{k} \theta##....?

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# Homework Help: Heat Conduction Equation - BVP

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