Question

find the general solution of the given system. (assume \\(\mathbf{x} = \

$$\begin{pmatrix} x \\\\ y \\end{pmatrix}$$

\\). enter any column vector as a row vector.)

\\\

$$\begin{aligned} \\frac{dx}{dt} &= 7x + 5y \\\\ \\frac{dy}{dt} &= -2x + 9y \\end{aligned}$$

\\

\\(\mathbf{x}(t) =\\)

Explanation:

Set up the system matrix

Using the Linear Systems of ODEs knowledge point

$$ \mathbf{A} =
LATEXBLOCK0
$$

Find the eigenvalues

Using the Linear Systems of ODEs knowledge point

$$
LATEXBLOCK1
$$

Find the eigenvector for \(\lambda = 8 + 3i\)

Using the Linear Systems of ODEs knowledge point

$$
LATEXBLOCK2
$$

Formulate the complex solution

Using the General Solution of ODE Systems knowledge point

$$
LATEXBLOCK3
$$

Extract real and imaginary parts

Using the General Solution of ODE Systems knowledge point

$$
LATEXBLOCK4
$$

Answer:

Find the general solution of the given system. (Assume \(\mathbf{X} =

$$\begin{pmatrix} x \\ y \end{pmatrix}$$

\). Enter any column vector as a row vector.)

\(\frac{dx}{dt} = 7x + 5y\)
\(\frac{dy}{dt} = -2x + 9y\)

\(\mathbf{X}(t) =\) <blank>\(c_1 e^{8t}

$$\begin{pmatrix} 5\cos(3t) \\ \cos(3t) - 3\sin(3t) \end{pmatrix}$$

+ c_2 e^{8t}

$$\begin{pmatrix} 5\sin(3t) \\ 3\cos(3t) + \sin(3t) \end{pmatrix}$$

\)</blank>