Laplace transform of integral equation

Question

Use Laplace transforms to solve the integral equation

$$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$

First find the Laplace transform $Y(s)$ of $y(t)$

Answer 1

Taking Laplace transform of the integral equation gives

$$ Y(s)-\frac{1}{2}\frac{Y(s)}{s}=\frac{1}{s}\implies Y(s) = \frac{2}{2s-1}. $$

Now, all you need to do is to find the inverse Laplace transform which will give you the solution

$$ y(t) = e^{t/2} . $$

Note: The integral

$$ \int_{0}^{t}y(t-v)dv $$

is the convolution of the functions $1$ and $y(t)$ and we used the fact

$$ \mathcal{L}(f*g) = \mathcal{L}(f)\mathcal{L}(g). $$