Question

4. (a) Expand the given function in an appropriate cosine or sine series. (x) , , -1<x<0 05x< (6 marks) (b) Find the product solutions for the given partial differential equation by using separation of variables. U, +3u, = 0 (6 marks)

Answer 1

The given fundion is f(x) = {2 -[LxLo g o 1-2 osx<T. Now, for the sake of continuity we define P(- TC) = (2) f (T) =(-2) Now, f is piece wise Monotone in [-1, n] . It is bounded and beriodic with period an. & is continuous on [-R, TI] except at x=0 Henu f is Integrable on [-1, n] . Thus f satisfies the dirichlet Condition on [-2, 0). Have & can be represented by fourier series which is nx + bn sinne) (x2 + (r e lolosna + I at E (an TI (-2) dx = 2 -2 =0 2 dx f(x) da til ... Now, - Noel - to "909.. - few of 44 23-2=0 our + Sw.coa de cat (20w it a befes kesen die 77 2 los nx dx tt [ (-2). Los na da I On f(x). Cosnx.da. - us plosions de seda * -filsanni di ( TL - -TU TU -- (1 - cos nt) + 2 (WsnN-1) - 4 (wsnr-1) = so if his even T- & if nis odd I Th

29 Thus the fourier series for f in [~R, T] is about a -8. [ Sinx Sin3x + Sin5x + ..... I (Ans) where X (X) Y(9) -3x Constant - 2x 6) The given partial differential equation is Uly +34y = 0 - 0 - Let u(x,y) = x(x). Y (Y) buting this in (1) we get x'(x). Y(Y) + 3 X(X). Y'Y) = 0 → x'(x). Y(Y) = - 3x(x). y'LY) = 3 ... Now, X'(x) = -37.X(*) ne s dx = -30% = dx = -32.dx. Integrating we get logx = - 32x + loger s log (4/4) = -32x S = X(X) = q. e Noco, y 'ly) = 2.YLY) - dy-ay soy ady - Integrating we get logy = xy + logez a Y (Y) - Ceny Therefore the general solution ist u(x,y) = Ge 32 gely - A e ?(y - 3x) A = G.2 is Hori 18 - analis Arbitary constant l and DER where