Find the differential equation that represents this following LTI system
I found
$$y(t) = \int \left( x(t) - 6 \int y(t) - 5 y(t)\right)$$
But after finding this differential equation of system, I need to find $y(t)$ for input $$x(t) = (e^{−t} + e^{−4t} )u(t)$$ by assuming system is at rest initially for this input. Should I also assume this initial rest situation for question part a.
You can now take the derivatives of this equation to get a differential equation, resulting in $$ y''+5y'+6y=x'. $$ You can avoid dealing with the derivative of $x$ by setting $v=y'-x$ so that \begin{align} y'&=x+v,\\ v'&=-5(v+x)-6y. \end{align} Or use the Laplace transform formalism to get to the solution.