Then I guess the question is "How good are you at differential equations". I can imagine having a son in secondary school that is working on partial differential equations can be rough! There is a standard method for solving the heat equation: separation of variables. If you let U(x,t)= X(x)T(t); that is, assume U can be written as the product of two functions, one, X, depending only on X and the other, T, depending only on t, then Uxx= X"T and Ut= XT' (' and " are first and second derivatives with respect to the appropriate variable). The equation becomes XT'= KX"T. Dividing through by XT gives
T'/T= K X"/X. That is, the left side depends only on t and the right only on x. If we vary x while t remains the same, the left side doesn't change- therefore the right side can't either: K X"/X must be a constant. Similarly, the left side must be that same constant for all t. Call that constant C. We now have the two ordinary differential equations T'/T= C or T'= CT and K X"/X= C or KX"= CX. If your son cannot solve those equations he should go back and study ordinary differential equations before he gets into partial differential equations! The first has the obvious solution T= AeCt. The solution to the second depends upon whether C is positive or negative. If C is positive then the solution will involve exponentials and will be unbounded at either + or - infinity so C must be negative. In that case, the solution will be in terms of sine and cosine or, equivalently, complex exponentials. In general you will need a sum (more correctly, an integral) of such things: the Fourier Transform.
Before doing a problem like this, your son should be comfortable with ordinary differential equations, partial differential equations over finite integrals, whose solutions involve Fourier series, and the beginnings of Fourier Transform theory. On the other hand he could just go outside and play soccer and wait for college.
