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Section 6.6 – The Convolution Integral

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. Find the following convolutions using the definition only.
    1. \(e^t*e^{3t}\)
    2. \(t*t^n\), where \(n=0, 1, 2, \dots\)​

      1. \(\dfrac{1}{2}e^{3t}-\dfrac{1}{2}e^t\)
      2. \(\dfrac{t^{n+2}}{(n+1)(n+2)}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V60


  2. Using the Laplace transform (instead of the definition), compute the following convolutions.
    1. \(u_a(t)*u_b(t)\)
    2. \(t^n*t^m,~~~ n, m=0, 1, 2, \dots\)​

      1. \(u_{a+b}(t)(t-(a+b)\)
      2. \(\dfrac{n!m!}{(n+m+1)!}t^{n+m+1}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V61


  3. In each of the following cases, find a function (or a generalized function) \(g(t)\) that satisfies the equality for \(t\ge 0\).
    1. \(t*g(t)=t^4\)
    2. \(1*1*g(t)=t^2\)
    3. \(1*g(t)=1\)​

      1. \(g(t)=12t^2\)
      2. \(g(t)=2\)
      3. \(g(t)=\delta(t)\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V62


  4. Write the inverse Laplace transform of the function \(F(s)=\dfrac{s}{(s+1)^2(s+4)^3}\) in terms of a convolution integral.

    \(f(t)=\displaystyle \int_{0}^t e^{-\tau}(\tau-1)\cdot\dfrac{1}{2}(t-\tau)^2\cdot e^{-4(t-\tau)} d\tau\)

    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V63


  5. Solve the initial value problem \[y''-2y'-3y=g(t), ~~~y(0)=1,~~y'(0)=-3\]

    \(y(t)=-\dfrac{1}{2}e^{3t}+\dfrac{3}{2}e^{-t}+\dfrac{1}{4}\displaystyle \int_{0}^tg(t-\tau)(e^{3\tau}-e^{-\tau}) d\tau\)


    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V64