Non-linear non-constant coefficient second order ODE

[Curious me]

I would like to solve the steady-state one dimensional heat equation for a two piece material system. The thermal conductivity in each segment is a linear function of temperature, where ##\kappa_1=a_1T+b_1## for material 1 and ##\kappa_2=a_2T+b_2## for material 2. ##a_1, a_2, b_1, and \;b_2## are constants and T is temperature. Essentially ##\kappa## depends on both temperature and space since we have two materials.

I will explain my approach to solving this. I am hoping to see what you think about its correctness and if you identify it as an approach with a specific name that I am unaware of. Thank you. I should specify that using the described method I have obtained an analytical solution in Mathematica.

First, solve ##\frac{\partial}{\partial x}(\kappa\frac{\partial T}{\partial x})=0##. This will give you a temperature profile as a function of x with two unknown coefficients that are to be determined by the boundary conditions.

Then substitute ##\kappa## with ##\kappa_1\; and\; \kappa_2## to obtain two separate temperature profiles for each section. Now you will have to determine four coefficients. To do this use the following boundary conditions:
##1)\; T_i=T_h\;##
##2)\; T_f=T_c## where ##T_h## and ##T_c## are known.
##3)\;\kappa_1\frac{\partial T_1}{\partial x}=\kappa_2\frac{\partial T_2}{\partial x}## (constant heat flux)
4) the boundary temperatures equate.

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Thank you

 

[gtoguy97]

for your explanation. Your approach is correct and is known as the method of separation of variables. This method involves separating the variables in the equation into individual terms, solving each term, and then combining them together to obtain the solution. In this case, you have separated the equation into two parts by substituting the thermal conductivity for each segment. You then used the boundary conditions to solve for the coefficients.