# representation for differential equation

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