Use Laplace transforms to solve the integral equation.

Question

enter image description here

I know I have to use convolutions but I'm not sure how. Any help is appreciated. https://i.stack.imgur.com/rFwnk.png

Answer 1

Note that $\int_0^t f(\tau) d \tau$ can be viewed as the convolution $1 * f$, so when you take the Laplace transform the equation becomes $$\frac{1}{s} g (s) + g(s) = \frac{1}{s^2}$$ where $g$ is the transform of $f$. Then $g(s) = \frac{1}{s(s+1)}$, so $f(t) = 1- e^{-t}$.

Answer 2

Hint.

Considering $L(\cdot)$ the Laplace transform operator

$$ L\left(\int_{\tau=0}^{\tau = t}y(\tau)g(t-\tau)d\tau\right)=L(y)L(g)=Y(s)G(s) $$

now assuming $g(t-\tau) = 1$ we obtain

$$ L\left(\int_{\tau=0}^{\tau = t}y(\tau)d\tau\right)=L(y)L(1)=Y(s)\frac 1s $$