Question

6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ Bk sin kx fx). (The fact that this series sums to the given initial temperature f(x) solves the so-called problem of Fourier). b) The one dimensional wave equation Usy Ut determines the vertical displacement y ofa string of length L at time t. Assuming L-T, and U(0, t)- UL, t)-0 for all t, show that for any positive integer k, the functions sin kx sin ckt are solutions and sin kx cos ckt c) It is evident that the functions UGo) -In r and U(0) - 0 both satisfy Laplace's equation in polar coordinates as stated in the last problem. Show that U(x, y)- In vx2+y2 and U(x, y) arctan y/x both satisfy Usx+ Usy 0 in cartesian coordinates.

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