I don't know how to go about solving these questions the second one seems to want integration by parts but I don't know how that will work out, an the first one seems to need the definition of Laplace transforms but I don't see how that will work out.
Show that for any function f(t) for which the Laplace transform $L [f(t)] = F(s)$ exists, $L[tf(t)] = −F'(s)$
Solve the equation for $x(t)$ : $$\int_0^{t}(x(τ )(t− τ ))dτ = t^4$$
You can solve this problem by using the property of convolution theorem.
Following are the steps to solve these type of problems:
First, take the Laplace transform on both the sides.
Then, on the left hand side apply convolution theorem in which $f(t)=x(t)$ and $g (t) = t$. After calculating $x(s)$, take the inverse transform.