a general solution is given by θ=(C1 cosΛx+C2sinΛx)(C3e^Λy+C4e^(-Λy)), where θ = (T(x,y) - T1)/(T2 - T1) where T1 and T2 are suitably chosen values, usually based on the boundary conditions. It is got by non - dimensionalizing and simplifying the 2-D conduction equation which reduces to a Laplace equation form for constant heat conductivity and no heat generation. A solution of the form θ= X(x) * Y(y) is initially assumed and simplified.

The specific solution can be obtained by applying boundary conditions and the principle of orthogonality of a function. For more information and the derivation, refer "Fundamentals of Heat and Mass Transfer", 7th edition by Frank Incropera et. al..