Given an equation for $Y(s)$ and $X(s)$ $$ \frac{Y(s)}{1-sY(s)} = \frac{1}{\omega^2} X(s) - \frac{1}{\omega}, $$ where $Y(s) = \int_0^\infty e^{-st}y(t)dt$, $X(s) = \int_0^\infty e^{-st}x(t)dt$ are Laplace transforms of dynamical variables $y(t)$ and $x(t)$. Initially, $y(0) =1$. I want to obtain the equation of $y(t)$ in terms of $x(t)$.
Here is what I did.
Solution: Rewrite the equation as $$ Y(s) = \frac{1}{\omega^2} X(s) [1-sY(s)] - \frac{1}{\omega} [1-sY(s)]. $$ Because $y(0)=1$, therefore, $$ Y(s) = -\frac{1}{\omega^2} X(s) [sY(s)-y(0)] + \frac{1}{\omega} [sY(s)-y(0)]. $$ Perform inverse Laplace transform for both sides of the above equation, $$ y(t) = -\frac{1}{\omega^2}\int_0^t d\tau \ x(t-\tau)\frac{dy(\tau)}{d\tau} +\frac{1}{\omega} \frac{dy(t)}{dt}. $$ Then I obtain the equation of $y(t)$ in terms of $x(t)$: $$ \frac{dy(t)}{dt} = \omega y(t) + \frac{1}{\omega}\int_0^t d\tau \ x(t-\tau)\frac{dy(\tau)}{d\tau}. $$ My question: I am not sure if the procedure or the final equation is correct. Could you please give suggestions about my solution or is there other methods to obtain the equation of $y(t)$?