I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!!
Show that if $A(t)$ is partitioned as
$$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ 0 & A_{22}(t) & \\ \end{pmatrix}, $$
where $A_{11}(t)$ and $A_{22}(t)$ are square, then F(t,tau) is the transition matrix of the system such that:
$$ F(t,\tau) = \begin{pmatrix} F_{11}(t,\tau) & F_{12}(t,\tau) \\ 0 & F_{22}(t,\tau) & \\ \end{pmatrix}, $$
where
$$ \frac{\partial}{\partial t}F_{jj}(t,\tau)= A_{jj}(t)F_{jj}(t,\tau), \: j=1,2 $$
Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?
We can write the dynamical equations as follows:
$\begin{align*} \dot{x}_1 (t) &= A_{11}(t) x_1(t) + A_{12}(t) x_2(t) \\ \dot{x}_2 (t) &= A_{22}(t) x_2(t) \end{align*}$
Then we can write the following solutions:
$\begin{align*} x_2(t) &= F_{22}(t, t_0) x_2(0) \\ x_1(t) &= F_{11}(t, t_0) x_1(0) + \int_{t_0}^t F_{11}(t, \tau) A_{12}(\tau) x_2(\tau) d\tau\\ &= F_{11}(t, t_0) x_1(0) + \left( \int_{t_0}^t F_{11}(t, \tau) A_{12}(\tau)F_{22}(\tau, t_0) d\tau \right) x_2(0) \end{align*}$
Remember the solutions to the linear differential equations. Then we can write the following:
$\begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} F_{11}(t, t_0) & F_{12}(t, t_0) \\ 0 & F_{22}(t, t_0) \end{pmatrix} \begin{pmatrix} x_1(t_0) \\ x_2(t_0) \end{pmatrix}$
where
$\displaystyle F_{12}(t, t_0) = \int_{t_0}^t F_{11}(t, \tau) A_{12}(\tau)F_{22}(\tau, t_0) d\tau$
Hence, $F(t, t_0)$ is the solution of the original dynamical equation and $F_{jj}(t, \tau)$ is defined as in the question.