Please consider the following theorem:
Theorem (Floquet): Consider the equation $\dot{x} = A(t)x$ with $A(t)$ a continuous $T$-periodic $n\times n$-matrix. Each fundamental matrix $\Phi(t)$ of the equation $\dot{x} = A(t)x$ can be written as the product of two $n\times n$-matrices $$ \Phi(t) = P(t)e^{Bt} $$ with $P(t)$ $T$-periodic and $B$ a constant $n\times n$-matrix.
I want to solve the following exercise:
Consider the equation $\dot{x} = A(t)x$ with $A(t)$ a smooth $T$-periodic $n\times n$-matrix, $x\in\mathbb{R}^n$, $f(t)$ a smooth scalar $T$-periodic function. Consider the case when $n = 1, A(t) = f(t)$. Determine $P(t)$ and $B$ in the Floquet theorem.
What I've tried: If $n = 1, A(t) = f(t)$ the fundamental matrix $\Phi(t) = x(t)$ is a scalar function and the solution to the equation $\dot{x} = f(t)x$. According to the Floquet theoren we can write $\Phi(t) = x(t) = P(t)e^{Bt} = p(t)e^{bt}$ where $p(t)$ is a $T$-periodic scalar function and $b$ a scalar. If I substitute this into the original equation I get the following $$ \dot{x} = f(t)p(t)e^{bt} $$ Unfortunately I don't really know how to proceed from here..
Question: How should I solve this exercise?