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I have to find solution of followong volterra equation

$y(x)= x - \int_{0}^{x}(x-t) y(t) dt$ with $y(0) = 0$.

My attempt:

I differentiated the above and got

$y^\prime = 1 - \int_{0}^{x} y(t) dt \tag 1$

Again differentiating I get

$y^{\prime \prime} = 0 - y(x)$. $~~~~~$Now we get its solution as :

$y = c_1 \cos x +c_2 \sin x$ and using the condition $y(0) = 0$ we get

$y = c_2 \sin x$. Now when we differentiate this $y$ and compare with equation (1) at $x = 0$, we get $c_2 = 1$ and this gives $y = \sin x $ as the answer. Is my solution correct? Kindly rectify if I am wrong. Thanks.

abel
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monalisa
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1 Answers1

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here is a way of verifying/deriving your solution starting from $$y^{\prime \prime} = -y \text{ and initial conditions }y(0) = 0, y^\prime(0) = 1.$$

$\begin{align} y(x) & = \int_0^x y^\prime(t)\ dt = \int_0^x y^\prime(t)d(t-x) \\ & = y^\prime(t)(t-x)|_0^x - \int_0^x (t-x)y^{\prime \prime}(t) \ dt \\ &= x y^\prime (0)+\int_0^x (t-x)y(t) dt \\ &=x - \int_0^x(x-t)y(t)\ dt \end{align}$

abel
  • 29,170