MEC702_Final_Exam_Revision_Questions

# MEC702_Final_Exam_Revision_Questions - MEC702 Final Exam...

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Unformatted text preview: MEC702 Final Exam Revision Questions Section A 1. Solve the ODE. Find a general solution. For IVPs, solve completely. (a) y3y0 + x3 = 0 (b) y0 = sec2 y (c) y0 sin 2πx = πy cos 2πx (d) xy0 = y2 + y, (Set xy = u) (e) xy0 = x + y, (Set xy = u) 2. Test for Exactness. If exact, then solve. If not, then use an integrating factor. If IVP, then solve completely. (a) sin x cos ydx + cos x sin ydy = 0. (b) e3θ (dr + 3rdθ) = 0. (c) (x2 + y2)dx − 2xydy = 0. (d) 3(y + 1)dx = 2xdy, (y + 1)x−4 3. Find the general solution for the linear ODEs. If IVP, then solve completely. (a) y0 − y = 5.2 (b) y0 = 2y − 4x (c) y0 + ky = e−kx (d) y0 + 2y = 4 cos 2x, y( 41 π) = 3 (e) xy0 = 2y + x3ex 1 Section B 1. Find the Wronskian. (a) e4x , e−1.5x . (b) e−0.4x , e−2.6x . (c) x3, x2 (d) x, x1 2. Solve the Euler-Cauchy Equations. (a) x2y00 − 4xy0 + 6y = 0, y(1) = 0.4, y0(1) = 0. (b) x2y00 + 3xy0 + 0.75y = 0, y(1) = 1, y0(1) = −1.5. (c) x2y00 + xy0 + 9y = 0, y(1) = 0, y0(1) = 2.5. (d) x2y00 + 3xy0 + y = 0. 3. Find a general solution y = yh + yp where yp is found using the Method of Undetermined Coecients. (a) y00 + 5y0 + 4y = 10e−3x . (b) 10y00 + 50y0 + 57.6y = cos x. (c) y00 + 3y0 + 2y = 12x2. (d) 3y00 + 27y = 2 cos x + cos 3x. 2 Section C 1. Find a general solution y = yh + yp where yp is found using the Method of Variation of Parameters. (a) y00 + 9y = sec 3x. (b) y00 + 9y = csc 3x. (c) y000 − 2y00 − 9y0 + 18y = e2x (d) yiv + 5y00 + 4y = 90 sin 4x (e) yiv − 5y00 + 4y = 10e−3x 2. Solve the IVPs by the Laplace transform. Show all details. (a) y0 + 5.2y = 1.4 sin 2t, y(0) = 0 (b) y0 + 2y = 0, y(0) = 1.5 (c) y00 − y0 − 6y = 0, y(0) = 11, y0(0) = 28 (d) y00 + 9y = 10e−t, y(0) = 0, y0(0) = 0 (e) y00 − 14 y = 0, y(0) = 12, y0(0) = 0 3 Section D 1. Determine the type of critical point and its stability as well. a.) y10 = y1 y20 = 2y2 c.) y10 = y2 y20 = −9y1 b.) y10 = −4y1 y20 = −3y2 d.) y10 = 2y1 + y2 y20 = 5y1 − 2y2 2. Solve the following initial value problems. a.) y10 = y1 + y2 , y20 y2 (0) = 6 y1 (0) = 7, y2 (0) = 7 y1 (0) = 7, y2 (0) = 2 = 4y1 + y2 b.) y10 = 3y1 + 2y2 , y20 y1 (0) = 1, = 2y1 + 3y2 c.) y10 = −y1 + 5y2 , y20 = −y1 + 3y2 3. Solve the given higher-order ODEs. (a) y000 + 25y0 = 0. (b) yiv + 2y00 + y = 0. (c) yiv − 4y00 = 0. (d) (D3 − D2 − D + 1)y = 0. 4 Section E 1. Determine if the given functions are odd or even or neither. Find the Fourier series for the function. Show details of your work. (a) (b) (c) (d) (e) ( −x, −1 < x < 0 p = 2L f (x) = |x| = +x, 0 < x < 1 ( 1, −2 < x < 0 p = 2L f (x) = −1, 0 < x < 2 ( x + L, −L < x < 0 p = 2L f (x) = L − x, 0 < x < L ( 0, −π < x < 0 p = 2L f (x) = 1, 0 < x < π ( 0, −0.5 < x < 0 p = 2L f (x) = x, 0 < x < 0.5 , period . , period , period , period , period . . . 2. Find the type, transform to normal form and solve. Show your work in details. (a) uxx − 64uyy = 0. Solution: u(x, y) = f1 (x + 8t) + f2(x − 8t) (b) uxx − 16uyy = 0. Solution: u(x, y) = f1 (x + 4t) + f2(x − 4t) 5 ...
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• Winter '16
• Alveen Chand
• Laplace, Cos, Periodic function, general solution, 2L

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