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Laplace Integral Equation

  • Thread starter Trestal
  • Start date
  • #1
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Homework Statement


By taking the Laplace transform and using the convolution theorem, obtain the solution of the integral equation


Homework Equations


f(t) = sin t + ∫e^(t-u)*f(u) du
integral is from 0 to t


The Attempt at a Solution


I used the following site as a reference for how to construct the problem
http://www.solitaryroad.com/c915.html

I rewrote the equation using the convolution theorem to be this
f(t) = sin t + e^t*f(t)
Letting y = L{f(t)} this becomes
y = 1/s^2 + y/s-1

The website that i referenced you too somehow removes the y and gets the RHS purely in terms of s. I cannot reproduce the simplication the site used on their problem nor can i apply it to my own. I get
y = y(s^2+1)+(s-2)/[(s^2+1)(s-2)]

Hopefully I am just missing something obvious but I am unsure what to do from here. I will continue to play around with it but hopefully someone can nudge me in the right direction.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
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What you are missing is basic algebra!

Solve y = 1/s^2 + y/(s-1) for y and apply the inverse transform.
 
  • #3
3
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Finally got it. Took me hours to work through that but I just couldn't see a solution until you gave me a push. Cheers

Solve for y then solve using partial fractions before being able to invert
Final answer
F(t) = 1/5*e^2t - 1/5cos(t) + 3/5sin(t)

Thanks again!
 

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