Why is the solution to a non-homogenous linear ODE written in terms of a general fundamental solution and not a matrix exponential? Generally, I see the solution to a non-homogenous linear ODE
$$ \dot{x} = Ax + b(t)\\ x(0) = x_0 $$
written as
$$ x(t) = \Phi(t)\Phi^{-1}(0)x_0 + \int_0^t \Phi(t)\Phi^{-1}(\tau)b(\tau)d\tau $$
where
$$ \Phi^\prime(t)=A\Phi(t). $$
At the same time, I thought $\Phi(t)=\mathbb{e}^{At}$, which means that $\Phi(0)=I$ and the above could at least be simplified to
$$ x(t) = \Phi(t)x_0 + \int_0^t \Phi(t)\Phi^{-1}(\tau)b(\tau)d\tau $$
if not further to
$$ x(t) = \mathbb{e}^{At}x_0 + \mathbb{e}^{At}\int_0^t (\mathbb{e}^{A\tau})^{-1}b(\tau)d\tau. $$
Why is it, then, that the non-homgenous solution is written as originally stated in terms of $\Phi$? Is there a change of basis or generalization that I'm not aware of, or is there a mistake in this simplication?
