MEC702_Final_Exam_Revision_Questions - MEC702 Final Exam...

Info icon This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 Coecients. (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 ...
View Full Document

  • Winter '16
  • Alveen Chand
  • Laplace, Cos, Periodic function, general solution, 2L

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern