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## Homework Statement

A thermos consists of two layers (see fig). The core contains a liquid with temperature [tex]T_1[/tex]. The outer layer contains air at low pressure with temperature [tex]T_2[/tex]. The surrounding air has temperature [tex]T_3[/tex]. At [tex]t=0[/tex], [tex]T_2=T_3[/tex]. Assume that [tex]T_3[/tex] does not change with time, and that temperature is conserved between the core and outer layer. What is the magnitude of [tex]T_2[/tex] after [tex]t[/tex] seconds?

http://img141.imageshack.us/img141/3128/termos.png [Broken]

## Homework Equations

Heat equation: [tex]\frac{dT}{dt}=k\Delta T[/tex]

## The Attempt at a Solution

First I write down the differential equations for each temperature and layer:

[tex]\frac{dT_1}{dt}=k_1(T_2-T_1)[/tex]

[tex]\frac{dT_2}{dt}_1=k_1(T_1-T_2)[/tex]

[tex]\frac{dT_2}{dt}_2=k_2(T_3-T_2)[/tex]

[tex]\frac{dT_2}{dt}=k_1(T_1-T_2)+k_2(T_3-T_2)[/tex]

[tex]\frac{dT_3}{dt}=0[/tex]

Since I have two functions to deal with, [tex]T_1[/tex] and [tex]T_2[/tex], I set up a set of differential equations with two functions:

[tex]\frac{dT_2}{dt}=-T_2(k_1+k_2)+k_1T_1+k_2T_3[/tex]

[tex]\frac{dT_1}{dt}=k_1(T_2-T_1)[/tex]

How do I solve this set of differential equations. I dont think I can solve them one by one, since neither [tex]T_1[/tex] or [tex]T_2[/tex] is constant.

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