To help cool down computer chips, heat sinks like the one shown below are often employed to carry away the heat generated more efficiently:

Consider one of the metal pins, represented in the following schematic diagram:

it temperature $T_a$

Its temperature profile can be described by the following partial differential equation $($PDE$)$:

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=(\frac{de{l}^{2}T}{del{x}^2})-(T-T_a)$

where the temperature $T(t,x)$ is a function of both time and location $($measured axially from the root of the pin$),$ is the thermal diffusivity that measures heat conduction in the metal, $$ is a parameter that measures heat convection from the metal pin to the surrounding air, and $T_a$ is the temperature of the air around the pin.

$($a$)$ Suppose we are only interested in the steady$-$state temperature profile of the pin, i$.$e$.,$ when the computer chip has been running continuously for a while, and ejects a constant flux of heat to the pin. The PDE can then be simplified to a second$-$order ODE for $T(x)$ :

$0=(\frac{d^{2}T}{d{x}^2})-(T-T_a)$

with the boundary conditions: