The exposed surface \((x=0)\) of a plane wall of thermal conductivity, \(k\), is subjected to microwave radiation that causes the heat generation rate within the wall to vary as:

\[\dot{q}(x)=\dot{q}_{o}\left(1-\frac{x}{L}\right)\]

The boundary at \(x=L\) is insulated while the wall at \(x=0\) is maintained at \(T=T_{o}\).

a. Derive the differential equation for the temperature profile.

b. What are the boundary conditions for this problem?

c. Solve the differential equation to obtain the temperature profile.