Solving ODE by integral factor rule

Question

I've been reading a paper where the authors present an equation which I here rewrite as: $$y(t) = \int_0^t\frac{dx(s)}{dt}\exp\left(-\int_s^t \frac{dz}{a(z)}\right)ds\tag{1}\label{1}$$ where $s$ and $z$ are dummy variables and the functions of time are continuous and continuously differentiable. I set my mind to get to this equation and I would like to know if the math I show is correct. I worked from the constitutive equations for the subject of the paper to obtain: $$\frac{dy(t)}{dt}+\frac{y(t)}{a(t)}=\frac{dx(t)}{dt}\tag{2}\label{2}$$ This equation is right and also shown in the paper and textbooks on the subject. I solve \eqref{2} by applying the integral factor rule, I assume I can because all functions are continuous; please correct me if I'm wrong. So if I call $I(t)$ the integral factor: $$I(t)=\exp\left(\int\frac{dt}{a(t)}\right)\tag{i}\label{i}$$ and I follow the method by multiplying \eqref{2} by $I(t)$ applying the product rule to the left side of the equation and integrating both sides so that I get: $$\int \frac {d}{dt}\left(y(t)\exp\left(\int\frac{dt}{a(t)}\right)\right)dt=\int\frac{dx(t)}{dt}\exp\left(\int\frac{dt}{a(t)}\right)dt\tag{3}\label{3}$$ and this can be rewritten as: $$y(t)\exp\left(\int\frac{dt}{a(t)}\right)+c=\int\frac{dx(t)}{dt}\exp\left(\int\frac{dt}{a(t)}\right)dt\tag{4}\label{4}$$ With $c$ an integration constant which we could consider $0$ given we are interested in relative evolution of $y(t)$ with time. At this point my main doubt is if it is right to write \eqref{3} as: $$y(t)\exp\left(\int_0^t\frac{ds}{a(s)}\right)=\int_0^t\frac{dx(s)}{dt}\exp\left(\int_0^s\frac{dz}{a(z)}\right)ds\tag{5}\label{5}$$ where I introduced $s$ and $z$ as dummy variables. If I set the integration boundary between $0$ and $t$ is it correct to write the left side in \eqref{5} the way I did? What is the rigorous way to go about it? My way of thinking is "the integral and the differential in the left side of \eqref{3} cancel each other out and since I integrate on the right between $0$ and $t$ then the integral in the left side exponential shares the same boundaries". From \eqref{5} I can then obtain \eqref{1} by considering: $$\exp\left(\int_0^s\frac{dz}{a(z)}\right)=\exp\left(\int_s^t\frac{-dz}{a(z)}\right)\exp\left(\int_0^t\frac{dz}{a(z)}\right)\tag{ii}\label{ii}$$ and for a given point $t$ the value $\exp\left(\int_0^t\frac{dz}{a(z)}\right)$ is a constant so I cancel from left and right of \eqref{5} and I obtain \eqref{1}. Do you think the math is right/clear? If you are interested in what this apply to, it's constitutive equations for viscoelastic materials models (in $1$D).