Do you understand the u' makes no sense here? You are not thinking in terms of more than one variable. u is a function of two variables and you must state whether you are differentiating with respect to x or t. Saying that the ends are insulated means there is no heat flow over them. "Heat flow", in one direction or another, is the change of temperature with respect to x. Your condition should be at the ends of the interval.
What does "D" represent? I suspect it is the "derivative operator" but, again, that makes no sense in a problem with two variables. You already have the derivatives with respect to x and t.For the equation i have u_t = Du_xx -x(L-x) but i am not sure whether i need the D at the front or not.