ENGR 9420 Engineering Analysis
Mid Term Test Questions

2006 Fall
[Ordinary Differential Equations; Perturbation Series]
  1. Find the complete solution to the initial value problem

    [10]

    (y^2 – y) dx + (xy – x) dy = 0 , y(1) = 1

  1. The response   x(t)   of a shock absorber to a sudden blow may be modelled by the initial value problem

    [10]

    x" + 2x' + 17x = 4 delta(t-5) , x(0) = x'(0) = 0

    where   d (t – a)   is the Dirac delta function.
    Using any valid method, find the response   x(t).
  1. Find a power series solution about   x = 0   of the ordinary differential equation

    [10]

    x^2 y" + 5x y' + 3y = 0

    using the method of Frobenius.
  1. Model the initial value problem

    [10]

                      y' + 2y = 0.01y^3, y(0)=1
    as a perturbation of the linear initial value problem
                      y' + 2y = 0, y(0)=1
    by finding the general solution of
                      y' + 2y = e y^3, y(0)=1
    as a perturbation series, as far as the term in e 2 and evaluating it at e = 0.01 .

  1. BONUS QUESTION:

    A rocket has a mass, when completely empty of fuel and payload, of   r.
    It carries a payload of mass   p.   Initially, it has a mass   f   of fuel aboard.
    Propulsion is provided purely by the expulsion of fuel from the rocket, at a constant rate of   a (kg/s) and a speed relative to the rocket of   u.
    [r, p, f, a and u   are all constants, measured in SI units.]

    1. Develop an ordinary differential equation that governs the speed   v(t)   of the rocket as it travels in free space.

      [+6]

    2. The rocket starts from rest.   Find its terminal speed (when the last of the fuel has been expelled) in terms of the constants   r, p, f, a and u.

      [+4]

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