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UNIVERSITY OF DUBLIN MA2E21 TRINITY COLLEGE Faculty of Science school of mathematics SF Engineering Trinity Term 2000 Course 2E2 Tuesday, June 6 Luce Hall 14.00 — 17.00 Dr. P. Tran Ngoc Bich Answer ALL questions 1. Solve the Initial Value Problem y 00 + 4 y 0 + 3 y = et + 2 δ(t) where y(0) = y 0 (0) = 0 . 2. Using the Second Shifting Theorem, find the Laplace transform of the function f (t) = (t − 1) H(t − 1) + H(t − 1) , where H is the Heaviside function. Solve the Initial Value Problem: y 00 − y = f (t) where y(0) = y 0 (0) = 0 . Calculate the first derivative of the solution y(t). 3. By means of the Z-transform, solve the difference equation un+2 − 3 un+1 + 2 un = 0 where u0 = 1 , u1 = −4 . 4. Using the matrix form of the Laplace transform, solve the system of differential equations x01 = 2 x1 − x2 + 3 x3 + (1 − t) , x02 = −x1 + x2 − x3 + (1 + t) , x03 = x2 − x3 + t . for the initial values: x1 (0) = x2 (0) = x3 (0) = 1. 2 MA2E21 5. Using the Power Series method, solve the differential equation y 00 − x y = 0 where y(0) = 1 , y 0 (0) = 0 . 6. Show that x0 = 1 is a regular singular point of the differential equation (1 − x2 ) y 00 − 2x y 0 + 2 y = 0 . Find the indicial equation of this differential equation. Deduce the general form of the solution (do not compute explicitly the solution). ~ · F~ and ∇ ~ × F~ for 7. Compute ∇ F~ = sin(x − z)~ı + y 2 z ~ + z 2 y ~k . 8. State Green’s theorem in the plane. Evaluate Z 2 2x cos(2y) dx − 2x sin(2y) dy C around the circle C of centre the origin and radius one. c UNIVERSITY OF DUBLIN 2000