Facebook Page
Twitter
RSS
Thanks Thanks:  0
+ Reply to Thread
Results 1 to 1 of 1
  1. MHB Master
    MHB Math Helper

    Status
    Offline
    Join Date
    Jan 2012
    Posts
    1,373
    Thanks
    130 times
    Thanked
    2,122 times
    Thank/Post
    1.546
    #1
    Quote Quote:
    $\displaystyle f\left( t \right)$ satisfies the integral equation

    $\displaystyle f\left( t \right) = 7\,t - 3\int_0^t{ f\left( u \right) \,\mathrm{e}^{-3\,\left( t - u \right) } \,\mathrm{d}u } $

    Find the solution to the integral equation using Laplace Transforms.
    This requires the convolution theorem:

    $\displaystyle \int_0^t{f\left( u \right) \,g\left( t- u \right) \,\mathrm{d}u } = F\left( s \right) \,G\left( s \right) $

    In this case, $\displaystyle g\left( t - u \right) = \mathrm{e}^{-3\,\left( t - u \right) } \implies g\left( t \right) = \mathrm{e}^{-3\,t } \implies G\left( s \right) = \frac{1}{s + 3}$.

    So upon taking the Laplace Transform of the integral equation, we have

    $\displaystyle \begin{align*} F\left( s \right) &= \frac{7}{s^2} - 3\,F\left( s \right) \left( \frac{1}{s + 3} \right) \\
    F\left( s \right) &= \frac{7}{s^2} - \frac{3\,F\left( s \right) }{s + 3} \\
    F\left( s \right) + \frac{3\,F\left( s \right) }{s + 3} &= \frac{7}{s^2} \\
    \left( 1 + \frac{3}{s + 3} \right) F\left( s \right) &= \frac{7}{s^2} \\
    \left( \frac{s + 6}{s + 3} \right) F\left( s \right) &= \frac{7}{s^2} \\
    F\left( s \right) &= \frac{7 \left( s + 3 \right) }{s^2\,\left( s + 6 \right) } \\
    F\left( s\right) &= \frac{7\,s + 21}{s^2\,\left( s + 6 \right) } \end{align*}$

    Taking the Inverse Transform will require Partial Fractions:

    $\displaystyle \begin{align*} \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s + 6} &\equiv \frac{7\,s + 21}{s^2\,\left( s + 6 \right) } \\
    A\,s\left( s + 6 \right) + B\,\left( s + 6 \right) + C\,s^2 &= 7\,s + 21 \end{align*}$

    Let $\displaystyle s = 0 \implies 6\,B = 21 \implies B = \frac{7}{2} $

    Let $\displaystyle s = -6 \implies 36\,C = -21 \implies C = -\frac{7}{12} $

    Thus $\displaystyle A\,s\left( s + 6 \right) + \frac{7}{2} \left( s + 6 \right) - \frac{7}{12}\,s^2 = 7\,s + 21 $.

    Let $\displaystyle s = 1 $

    $\displaystyle \begin{align*} 7\,A + \frac{7}{2} \cdot 7 - \frac{7}{12} \cdot 1^2 &= 7\cdot 7 + 21 \\
    7\,A + \frac{49}{2} - \frac{7}{12} &= 70 \\
    7\,A + \frac{294}{12} - \frac{7}{12} &= \frac{840}{12} \\
    7\,A + \frac{287}{12} &= \frac{840}{12} \\
    7\,A &= \frac{553}{12} \\
    A &= \frac{79}{12} \end{align*}$

    $\displaystyle \begin{align*} F\left( s \right) &= \frac{79}{12} \left( \frac{1}{s} \right) + \frac{7}{2} \left( \frac{1}{s^2} \right) - \frac{7}{12} \left( \frac{1}{s + 6} \right) \\
    f\left( t \right) &= \frac{79}{12} + \frac{7}{2}\,t - \frac{7}{12} \,\mathrm{e}^{-6\,t} \end{align*}$

  2. # ADS
    Circuit advertisement
    Join Date
    Always
    Posts
    Many
     

Similar Threads

  1. Mahesh's question via email about Laplace Transforms (1)
    By Prove It in forum Questions from Other Sites
    Replies: 0
    Last Post: March 24th, 2020, 23:03
  2. Alexander's question via email about Laplace Transforms
    By Prove It in forum Questions from Other Sites
    Replies: 0
    Last Post: March 22nd, 2020, 20:08
  3. Collin's question via email about solving a DE using Laplace Transforms
    By Prove It in forum Questions from Other Sites
    Replies: 0
    Last Post: August 12th, 2016, 10:49
  4. Collin's questions via email about Inverse Laplace Transforms
    By Prove It in forum Questions from Other Sites
    Replies: 0
    Last Post: August 12th, 2016, 10:09
  5. Douglas' question regarding Laplace Transforms (1)
    By Prove It in forum Questions from Other Sites
    Replies: 0
    Last Post: March 14th, 2014, 00:25

Tags for this Thread

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •  
Math Help Boards