Interaction Picture Propagator

From Page.81 of Peskin and Schroeder

… we note that $U(t,t_0)$ is the unique solution, with initial condition $U(t_0,t_0) = 1$, of
a simple differential equation(the Schrodinger equation):

$ i frac{partial}{partial t} U(t,t_0) = e^{iH_0(t-t_0)}(H-H_0)e^{-iH(t-t_0)}….$

How did the textbook arrive at this?

I have used the Heisenberg equation of motion:

$i frac{partial}{partial t} mathscr{O} = [mathscr{O},mathscr{H}]$

and arrived at the following point and am stuck:

$i frac{partial}{partial t} U(t,t_0) = [e^{iH_0(t-t_0)},H].e^{-iH(t-t_0)}$