1. The problem statement, all variables and given/known data In each of Problems 1 through 8 find the steady-state solution of the heat conduction equation ∂2uxx=ut that satisfieds that given set of boundary problems. ..... 3. ux(0,t)=0, u(L,t)=0 2. Relevant equations Assume u(x,t)=X(x)T(t) 3. The attempt at a solution ∂2X'' T = X T' X'' / X = (1/∂2) T' / T = -ß X'' + ßX = 0, T + ∂2ßT' = 0. Since ux(0,t)=0 and u(x,t)=X(x)T(t), X'(0)T(t)=0 ---> X'(0)=0. Solving X'' + ßX = 0 like a I would any differential equation, X(x) = C1cos(√(ß)x) + C2sin(√(ß)x) ---> X'(x) = -√(ß)C1sin(√(ß)x) + √(ß)C2(cos(√(ß)x) ---> X'(0)=0= 0 + √(ß)C2 ---> C2=0 ---> X(x) = C1cos(√(ß)x). Also, u(L,t)=0 ---> X(L)T(t)=0 ---> X(L)=0. I'm getting somewhere, right? Since we're talking about a steady-state solution being reached, some function of t and possibly x will disappear, leaving us with just a function v(x) that shows the temperature at any place in the rod. u(x,t) = v(x) + w(x,t), limt-->∞ u(x,t) = v(x) ............... Anyhow, I'm sort of stuck. Problem 1 was was easy because it gave me u(0,t)=T1, u(L,t)=T2, just like the examples in the chapter; but this one is throwing me off with the ux(0,t). I'm having trouble putting the pieces together. Please help, geniuses.