Finding State Transition Matrix

Can someone help me compute the State Transition Matrix for the following system of linear differential equations?

$$\vec x=\begin{bmatrix}2 & -t\\-t & 2\end{bmatrix} \vec x$$

My attempt:

Compute $A(t) = \begin{bmatrix}2 & -t\\-t & 2\end{bmatrix}$ so $\int A(t)=\begin{bmatrix}2t & \frac{-t^2}{2}\\ \frac{-t^2}{2}& 2t\end{bmatrix}$

Then I checked the equality of multiplications: $$A(t)\cdot\int A(t)d(t) =\int A(t) d(t) \cdot A(t)$$ and indeed they are equal.

Then I calculated the Eigenvalues as:

$$\lambda=\frac{4t\pm t^2}{2}$$

Plugging the Eigenvalues into the matrix:

$$\alpha \begin{bmatrix}1 \\1 \end{bmatrix}$$ and $$\beta \begin{bmatrix}1 \\-1 \end{bmatrix}$$

which gives $$H=\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$$

How do I proceed from here?

Solution 1:

Hint: Take $HAH^{-1}$ and then review your solution