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Problem 2: (25 points) Consider the following system of differential equations: $\frac{dx_1}{dt} = -x_1 + \frac{3}{2}x_2$ $\frac{dx_2}{dt} = -\frac{1}{6}x_1 - 2x_2$ $\vec{x}(2) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, were, $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$. a) Convert the given system to matrix form $\dot{X} = AX$? b) Find the eigenvalues and eigenvectors for the matrix A? Hint: For the eigenvectors, consider the first equation of the eigenvectors solution and assume: $\Rightarrow \eta_2 = 1 \quad \rho_2 = 0$ c) Using the results from part b), Find the general solution x(t) for the system? d) Applying the initial condition, find the actual solution x(t)?

Name: problem 2 25 points consider the following system of differential equations fracdx1dt x1 frac32x2 fracdx2dt frac16x1 2x2 vecx2 beginbmatrix 1 0 endbmatrix were vecx beginbmatrix x1 x 60579
Uploaded: 2023-09-23T04:12:14-08:00
Duration: 3 min 22 s
Channel: Sri K
Description: problem 2 25 points consider the following system of differential equations fracdx1dt x1 frac32x2 fracdx2dt frac16x1 2x2 vecx2 beginbmatrix 1 0 endbmatrix were vecx beginbmatrix x1 x 60579

          Problem 2: (25 points) Consider the following system of differential equations:
$\frac{dx_1}{dt} = -x_1 + \frac{3}{2}x_2$
$\frac{dx_2}{dt} = -\frac{1}{6}x_1 - 2x_2$
$\vec{x}(2) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, were, $\vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$.
a) Convert the given system to matrix form $\dot{X} = AX$?
b) Find the eigenvalues and eigenvectors for the matrix A? Hint: For the eigenvectors, consider the
first equation of the eigenvectors solution and assume:
$\Rightarrow \eta_2 = 1 \quad \rho_2 = 0$
c) Using the results from part b), Find the general solution x(t) for the system?
d) Applying the initial condition, find the actual solution x(t)?

$problem 2 25 points consider the following system of differential equations fracdx1dt x1 frac32x2 fracdx2dt frac16x1 2x2 vecx2 beginbmatrix 1 0 endbmatrix were vecx beginbmatrix x1 x 60579$

Added by David E.

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