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Consider the differential equation ut =1/2 uxx + ux for 0 <x < pi, t > 0 with boundary conditions u(0,t) = u(pi,t) = 0.
(a) Separate variables and write the ordinary differential equations that the space factor X(x) and the time factor T(t) must satisfy.
(b) Show that 0 is not an eigenvalue of the Sturm-Liouville problem for X.
(c) Show that for any integer n>1, Xn(x) = e^-x sinnx is an eigenfunction of the Sturm-Liouville problem for X and determine the corresponding eigenvalue.
(d) Assuming that these are all the eigenvalues, write down in series form the general solution of the boundary value problem above assuming a general initial condition u(x, 0) = f(x).
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