Fundamental matrices Find the fundamental matrix for the two-dimensional system defined by
$x_1' = x_1 + tx_2$, and  $x_2'=x_2$. And determine the solution for which $x_1(0)=c_1$, and $x_2(0)=c_2$.
I am stuck because of the '$t$' in the first equation, otherwise I would be able to do it with the eigenvalue method. I am not sure what to do in this case.
 A: We are asked to find the fundamental matrix for the two-dimensional system defined by:
$$\tag 1 x_1' = x_1 + t~x_2$$
$$\tag 2 x_2' = x_2$$ 
And determine the solution for which $x_1(0)=c_1$, and $x_2(0)=c_2$.
We note that these equations are both first-order linear and are decoupled. We are going to use this to our advantage, by solving for $x_2$ in $(2)$ and then substituting and solving $(1)$.
Solving for $(2)$ using integration yields:
$$x_2(t) = c_1e^t$$
Substituting this into $(1)$ yields $x'_1 = x_1 + c_1~t~e^t$. Solving for the homogeneous and non-homogeneous solutions yields:
$$x_1(t) = c_2~e^t + \dfrac{1}{2}~c_1~t^2~e^t $$
We want to find the fundamental matrix, but you did not specify how you typically show that, so I am going to show both. 
We are going to use initial conditions using $e_1 = (1, 0)$ and $e_2 = (0, 1)$. What theory allows you to use this (I will assume you learned that in class)?
For $e_1$, we have:
$$x_a = \begin{bmatrix}x_1(0)\\x_2(0)\end{bmatrix} = \begin{bmatrix}c_2\\c_1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix} \rightarrow c_1 = 0, c_2 = 1$$
This yields a solution:
$$x_a(t) = \begin{bmatrix}e^t\\0\end{bmatrix}$$
For $e_2$, we have:
$$x_b = \begin{bmatrix}x_1(0)\\x_2(0)\end{bmatrix} = \begin{bmatrix}c_2\\c_1\end{bmatrix} = \begin{bmatrix}0\\1\end{bmatrix} \rightarrow c_1 = 1, c_2 = 0$$
This yields a solution:
$$x_b(t) = \begin{bmatrix}\dfrac{1}{2}~t^2~e^t\\e^t\end{bmatrix}$$
Now, we can form the fundamental matrix using a linear combination of the solutions $x_a(t)$ and $x_b(t)$ (note, you should verify that both $x_a(t)$ and $x_b(t)$ each satisfy the original system) as:
$$\phi(t) = [x_a(t) | x_b(t)] = \begin{bmatrix}e^t & ~~\dfrac{1}{2}~t^2~e^t \\ 0 & e^t\end{bmatrix}$$
Note, you should verify that these solutions are linearly independent (how can you show that), thus their linear combination is also a solution of the system.
Sometimes, the fundamental matrix is given in the form $\phi(t, t_0)$, so I will include that one too. To find this variant, we can use the fundamental matrix we already found and calculate:
$$\phi(t, t_0) = \phi(t) \cdot \phi^{-1}(t_0) = \dfrac{e^t}{2} \begin{bmatrix}2e^{-t_0} & ~~e^{-t_0}(t^2 - t^2_0) \\ 0 & 2e^{-t_0}\end{bmatrix}$$
Note, you should verify that both of these fundamental matrices satisfy all of the required properties. For example $\phi(t, t) = I$. Also, that each of these actually satisfies the original system.
Lastly, there are many other ways to find the fundamental matrix and you should explore those. 
