If the characteristic equation for a differential equation can be written as $(s-r_1)(s-r_2)$, the substition $z=y'-r_1y$ yields an equation of the form $z'-r_2z=f(x)$.
For example, if our equation is
At this point, integrating factors can be used to solve for $z$, then substituting back will yield a solution for $y$. My first question is why does it work like this?
My second question is if there's a way to find a substitution for a general second order linear ODE
that will similarly reduce the problem to a first order linear ODE which can then be solved by integrating factors?