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please help with question 1
(5) Question 1 [10] (a) A rod of length L = 7 units, initially at a temperature u(x,0) = sin x, is thermally insulated at the ends. Inside the rod the temperature u varies in accordance with the equation Up = uzr. Find the temperature u(x, t) for t > 0 and 0 < x <T. (b) Find the power series solution of the form y(x) = Enzo Anxu” of the differential equation dy dc Question 2 [10] (a) Show that 1 dr = VT. 4 - ແy - -16% e = (b) Find the Maclaurin series of the function (3) $ sinh 20 dw. W = (4) (e) Find the Laurent expansion of f(z) = (22+1), in the region 2 < 121 <0. 22+) Question 3 [10] (a) Find the residue of the function f(z) = 44922 at all its singular points. (b) Using residue theorem, evaluate the integral = C G lo 27 9 cos 70 dᎾ . 10 - 6 cos e