Question

Differential Equations for Engineers II Page 3 of 6 3. The interface y(x) between air and water in a time-independent open channel flow can be approximated with the second order ODE dạy ta’y = 0, d.r2 >0, (3) 4 marks where the parameter a’ is a measure of the mean speed of the flow. The flow is in the positive x direction (i.e. from left to right). (a) The point x = 0 is an ordinary point of equation (3). Therefore, a power series solution of the form, y = Σ m=0 3 marks can be used to find the general solution to equation (3). Substitute the power series (and its derivatives) into equation (3) to determine the recurrence relation for the coefficients am+2 in terms of am. (b) Determine the coefficients az, az, 24 and az in terms of the coefficients a, and a1, and express your answer in the form y = 209(2) +azh(). (c) Show that your series solution is equivalent to y = A cos(ax) + B sin(ar) by determining A and B in terms of ao, aj and a. 1 mark

Answer 1

Ans d²y +2²y = 0 dx2 1a) y = E Am am mzo 2 dy É mlm-1) Am Xm-2 dx2 m=2 22 E Am x -0 E m/m-1) Am Xm-2 + mzo m=2 v for this. m-2=n. =0 E (n+2)(n+1) Antaxon + E 2² Amam mo no ✓ for this men m ir mo m+2 )(m+1) Omp3 +2? Am ]X (m+2) (M+1) Om +2 +2² am = 0 * 1b). for m=0 for 2X1 x A2 + 22 Mo = 0 -2² ao 02 - - 0 2 for m=1 +2²a = 0 3 x 2 +Az 22 ai 6 2 Az =

for m=2 for * 4X3XQ4 + 2²2 = 0 2 2²02 A4 = 12 By eq A4 = 2. 2 24 ao ao X 12 24 for m=3 5 x 4 As + 2²03 = 0 As = - Bar = 2 x 1-20) - 24 a, 120 y = ao + Q x + (-2.00) x ² = 220 x 3 + 24 Go xA 6 24 Aai x5+ t 20 23 2?x2 x² + 24 x 4 y = 0o[1 - 2 Ira, [x-2 2 x + 2 n 6 120 q 24 y = ao g(x) ta, h(x) 2 2 + C Lx 4! g(x) = 1-txt h(x) = x- 1 2 3 2 x 3! +24 x 5 + 5: Tinda g(x) = cos (2x) hao. y = Cocos (2x) + al Sin (226) B = ai sans L A = Mo
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