Investigate uniqueness of the differential equation

Thread starter ahmed39399
Start date Mar 20, 2010
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Differential Differential equation Uniqueness

Mar 20, 2010

ahmed39399

Investigate uniqueness of the differential equation

the question is in the image attached

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Mar 20, 2010

g_edgar

Since you have posted in the non-homework forum, perhaps you should begin by telling us your thoughts. What have you done and why did you get stuck?

Mar 20, 2010

ahmed39399

Really I want something to begin with>

I want a hint to begin

Thread 'Volterra equation of the second kind'

There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...

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