So, I am doing a project on Bessel Functions and one of the questions is:
Solve the following Volterra Integral Equation of the First Kind.
$$ \int_{0}^{x} J_{0}(x-t)y(t) dt = sin(x) $$
where, $J_{0}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(k!)^2} (\frac{x}{2})^{2k}$ is the Bessel Function of the First Kind.
Now, I know
$$ L(sin(t))(s) = \frac{1}{s^{2}+1} $$
But I am struggling to workout the integral with a summation and thus solve the equation.
Well, we have the following equality:
$$\int_0^x\text{J}_0\left(x-t\right)\text{y}\left(t\right)\space\text{d}t=\sin\left(x\right)\tag1$$
Now, when we want to take the Laplace transform we need to use the time-domain integration property of the Laplace transform:
$$\frac{\text{Y}\left(\text{s}\right)}{\sqrt{1+\text{s}^2}}=\frac{1}{1+\text{s}^2}\tag2$$
So:
$$\text{Y}\left(\text{s}\right)=\frac{\sqrt{1+\text{s}^2}}{1+\text{s}^2}=\frac{1}{\sqrt{1+\text{s}^2}}\tag3$$
Which gives:
$$\text{y}\left(t\right)=\text{J}_0\left(t\right)\tag4$$