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let be the differential equation

$$ -y''(x)+q(x)y(x)=0 $$ (1)

from the Cauchy method to solve differenial equations

$ -y''(x)+y(x)=f(x) $ so $ y(s)= \int_{0}^{s}dt \sin (s-t)f(s) $

then for the equation (1) is it valid the integral equation representation

$$ y(s)=\int_{0}^{s}dt \sin (s-t)q(s)y(s) $$

if we imposoe the asymptotic condtion $ y(x)=e^{iax} $as $ x \to \infty $ then the integral equation becomes (approximately)

$$ e^{ias} \approx \int_{0}^{s}dt \sin (s-t)q(x)y(s) $ $$

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What exactly are you asking us here? – Ron Gordon Feb 22 '13 at 18:20
to see if i can realte an ODE of second order to a volterra integral equation – Jose Garcia Feb 22 '13 at 18:33

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