Answer to Question #295988 in Differential Equations for Aswin

Question

An insulated rod of length has its ends A and B maintained at O^ C and 100 degrees * C respectively until steady state conditions prevailFind the steady -state temperature of the rod

Answer 1

Let us remind the heat transfer equation :

$\partial_t T = a \partial_{xx}T$ as the problem has only one spatial dimension.

The steady state corresponds to the condition $\partial_t T = 0; \; T(t,x)=T(x)$. Using this gives us

$\frac{d^2}{dx^2} T = 0$

And the general solution of such equation is given by

$T = C_1x + C_2$

Let us place the origin at the point $A$ and let us denote the rod's length by $L$. Then we have :

$C_2 = 0^\circ C$

$C_1L+0 = 100 \rightarrow C_1 = 100/L$

Therefore, the temperature is given by $T(x) = \frac{x}{L}\cdot 100$