I'm working on expressing formulas for ODE's, but I can't seem to understand why, that under the assumption that we have constant coefficient the formula
\begin{align*} y&=\frac{1}{\mu(t)}\bigg(\int\mu(t)b(t) dt +c\bigg)=\exp\bigg(\int a(t)dt\bigg)\bigg(\int\mu(t)b(t)dt+c\bigg) \end{align*} becomes or is deducible to \begin{align*} y&=e^{ta}c+\int e^{ta}z(t)dt. \end{align*}
If $a$ is constant, then $\int_0^t a\,ds = at$ so that $μ=\exp(-\int a(s)ds)=e^{-at}$. Inserting into the first formula gives $$ y(t)=ce^{at}+\int_0^te^{a(t-s)}b(s)ds. $$ You need to be careful to not mix integration variables, avoid labeling them with the same letter if the are in the same term.