21-120 2² ах2 ди = 2 +u Where u (x, 0) = 6e-3x əx at f(x) = { if0 0 + (1-x) (f) Prove that Fourier sine integral for f(x) = e-Bx, is given by (g) Find Fourier cosine transform of the function; (a) Classify and state the order of the equation (b) Evaluate √xe-³√xdx (c) Use concept of Gamma function to find I'(x-2), given [(x + 1) = 2² (d) Find the Directional derivative of f(x, y, z) = x²y z in the direction of 4i-3k at point p(1,-1,1) (e) Using the method of separation of variables, solve eaz ✓Question Two (a) Use Gamma or Beta function to evaluate (b) e(zax-x²) dx = VTT 2 2²4 Əy² 24 ду SECTION B (40 MARKS) = 0 200 wsin wx π 0 3²+w² p²√22p-p² dp dw (2 Marks) (3 Marks) (2 Marks) (2 Marks) (h) Find xydx where, on C, x and y are given in terms of parameters t by x = 3t² and y = t³ - 1 for t varying from 0 to 1 (2 Marks) (3 Marks) (3 Marks) (3 Marks) (5 Marks) (5 Marks) Question Three A metal bar of 10 cm long is insulated along its sides and the lateral surface. The ends are placed into ice of 0°C. Find Temperature of any point x on the bar at any time t if the initial (10 Marks) temperature is (2x²)°C 41 2 Question Four (a) Using the concept of Fourier integral representation show that: if x < 0 if x = 0 00 cos wx+W sin wx Jo 1+w2 dw = (b) Find Fourier cosine transform of the function and write your answer in sinc form; if 0 < x 0 Question Six (a) Find the solution of PDEs that satisfies the following condition for all 0 < x < π at t > 0 a²u(x,t) Əx² 9 (a) Compute ff(x³y² + cosmx+ sin ny)dA over the rectangle R = [-2, -1] x [0,1] (4 Marks) X a²u(x,t) at² u(0, t) = u(π, t) = 0 for all t > 0 u(x, 0) = x(π-x) for all 0 < x < TI u₂(x, 0) = 0 for all 0 < x < T (b) Classify uxx + 3uy-2ux + 24u, +5u = 0 as hyperbolic, parabola or elliptic Question seven (a) Use the concept of Gamma function to evaluate (6 Marks) Pdp -5 Jo z 3 e-%dz (b) Compute the following integral (x² + y² + z²) dzdydx (4 Marks) (3 Marks) (3 Marks) (7 Marks) (3 Marks) (5 Marks) (5 Marks)