Derivative, Diagonalizability, Jordan Canonical Form

Ex 1

\(\frac{dy}{dx}=2y\)

\(y=e^{2x}\)

Ex 2

\(\frac{dy_1}{dx}=y_1, \frac{dy_2}{dx}=2y_2\)

Thus,

\(y_1=e^x, y_2=e^{2x}\)

Ex 3

\(\frac{dy_1}{dx}=6y_1+10y_2,\frac{dy_2}{dx}=-2y_1-3y_2\)

Thus,

\[\frac{d}{dx}\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} =\begin{bmatrix} 6 & 10 \\ -2 & -3 \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \]