ENGI 2422 Engineering Mathematics 2

Faculty of Engineering and Applied Science
2008 Winter

Problem Set 10 Questions

[Laplace transforms, initial value problems]

  1. Where possible, use Laplace transform methods to solve the initial value problems in Problem Set 9.   The solutions will appear in another web page.

  1. Find the Laplace transform   F (s)   for each of the following functions   f (t) :

    1. f (t)   =   t4 e–2t
    2. f(t) = {t , 0<=t<1 ; exp(1-t) , t>=1}
    3. f (t)   =   4 sin23t
    4. f (t)   =   e–t d(t – 2)
    5. f (t)   =   t e–2t cos 3t

  1. Find the inverse Laplace transform   f (t)   for each of the following functions   F (s) :

    1. F(s) = (s + 4) / (s^2 + 16)
    2. F(s) = ((s+1)^2 - 16) / ((s+1)^2 + 16)^2
    3. F(s) = (6s^3 + 28s^2 + 21s + 26) / (s^4 + 4s^3 + 13s^2)
    4. F(s) = (1 + (s-1)*exp(-s)) / s

  1. Find the complete solution of the following initial value problems:

    1. y'' - 3 y' - 10 y = 1 ; y(0) = y'(0) = 1
    2. y''' + 3 y'' + 3 y' + y = e^(-t) ; y(0) = y'(0) = y''(0) = 0
    3. y'' + 2 y' + 5 y = {sin 2t, 0<=t<pi ; 0 else} ; y(0) = y'(0) = 0

  1. Find the complete solution to the integral equation

    y(t) = cos t + Integral_0^t {y(t-z) exp(-2z) dz}

  1. Find the Laplace transform   F (s)   for each of the following periodic functions   f (t) :

    1. f(t) = {2t, 0<=t<=2 ; 2*(4-t), 2<t<4} f(t+4) = f(t) for all t
    2. f(t) = {t, 0<=t<1 ; (t-2)^2, 1<=t<2} f(t+2) = f(t) for all t
    3. f (t)   =   | sin wt |       (which is the fully rectified sine function).

  1. Use Laplace transforms to show that the homogeneous differential equation
                    y" – (m + n) y' + mn y = 0
    with initial conditions         y(0) = a ,     y'(0) = b
    (where   m, n, a and b   are all constants and   m, n   are distinct)
    has the complete solution

    y(x) = [(b-an)*exp(mx) - (b-am)*exp(nx)] / (m-n)

  1. Find the complete solution to the initial value problem

    y" + 5y' + 6y = exp(-2t) ; y(0) = 0, y'(0) = 1

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      Created 2001 03 14 and most recently modified 2007 12 23 by Dr. G.H. George