What is the Correct Solution to a Second-Order Integral Equation?

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I'm an Aircraft Engineering student from England and stumbled across this forum while I was struggling with some second-order integral equation today. I came across an unsolved post that pushed me in the right direction towards an eventual correct answer. To return the favour I created an account to post my results back to the OP to correct him and show him where he was going wrong.. However, the post is a couple of years old and commenting has been disabled. Thanks anyway as you guys really helped me to overcome that problem.
 
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