ENGI 9420 Engineering Analysis

Faculty of Engineering and Applied Science
2007 Fall

Final Examination Questions

  1. Use the method of Gaussian elimination to solve the linear system

    [ 1 1 1 | 0 ] [ 2 3 2 | 1 ] [ 3 2 -2 | 5 ] [ 6 6 1 | 6 ]

    [16]

  1. A pair of simultaneous first order ODEs is defined by

    dx/dt = y ; dy/dt = x^2 + 4x

    1. Find both critical points of this system of first order ODEs.

      [3]

    2. Determine the nature and stability of both critical points.   In particular, provide a reason why it is likely that one of the critical points of the non-linear system is a centre (not a focus).

      [6]

    3. Sketch the phase portrait for the linear approximation to the non-linear system in the neighbourhoods of both critical points.

      [6]

    4. Hence sketch the phase portrait for the original non-linear system.

      [5]

    BONUS QUESTION

    1. Find the equation of the separatrix and add it to your sketch in part (d).   Note on your sketch the exact values of the x axis intercepts for the separatrix.

      [+6]

  1. A function   f (x)   is defined by

    f(x) = x^3 + e^x

    1. Show that   f ' (x) > 0   for all x.
      It then follows that   f (x) = 0   has a unique solution.

      [3]

    2. Use Newton’s method to find the solution to   f (x) = 0   correct to four decimal places.
      Justify your choice of an initial guess   x0.

      [13]

  1. Find the path   y = f (x)   between the points (0, 1) and (1, 1) for which the integral

    I = Integral[0 to 1] { 2x + 3y + (y')^2 } dx

    has an extremum and determine whether the extremum is a maximum or a minimum.

    [16]

  1. Find the Fourier series expansion of the function

    f(x) = 4 – x^2 , (abs(x) <= 2)

    [16]

  1. An arbitrary purely radial vector field F may be defined in spherical polar coordinates by F(r) = f(r) r^, where   f (r)   is some differentiable function of the distance   r   from the origin.

    1. Show that any purely radial vector field is irrotational (that is, curl F identically= vector 0).

      [4]

    2. Show that, in order for the divergence of F to be zero everywhere,   f (r)   must be inversely proportional to the square of the distance from the origin.

      [12]

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        Created 2007 12 13 and most recently modified 2007 12 13 by Dr. G.H. George.