1. The problem statement, all variables and given/known data I need to go into this test with great aplomb. In each of Problems 1 through 8 find the steady-state solution of the heat conduction equation a2uxx=ut that satisfies the given boundary conditions. 1. u(0,t)=10, u(50,t)=40 .... 3. ux(0,t)=0, u(L,t)=0 2. Relevant equations Will be using separation of variables; so assume u(x,t)=X(x)T(t) 3. The attempt at a solution After a long time---that is, as t approaches ∞---I anticipate that a steady state temperature distribution v(x) will be reached. Then u(x,t) will just be a2v''(x)=v'(t). Since v is not a function of t, v'(t) = 0 and v''(x)=0. v(x) must be a 1st degree polynomial: v(x) = Ax + B. Notice that the initial conditions to problem one imply that X(0)T(t)=10, X(50)T(t)=40. I don't want T(t) to be something trivial, so X(0)=10, X(50)=40 ----> 10= 0 + B ---> B = 10 40=A*50 + 10 ----> A = (40-30) / 50 = 1/5 -----> v(x) = x/5 + 10 That's the steady state heat distribution, which is all the problem asked for. Now problem 3. Notice the sub x. Hmmmm..... need to put the pieces of this puzzle together. ux(0,t)=0 is saying that the rod is insulated at x=0. So X'(0)=0. The other initial condition says X(50)=40. Where do I go from here?