Question

use the method of undetermined coefficients to solve the given nonhomogeneous system. (assume \\(\mathbf{x} = \

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

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

\\\

$$\begin{aligned} \\frac{dx}{dt} &= 2x + 3y - 7 \\\\ \\frac{dy}{dt} &= -x - 2y + 6 \\end{aligned}$$

\\

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

Explanation:

Find eigenvalues of the coefficient matrix

Using the Linear Systems of ODEs knowledge point

$$ A =
LATEXBLOCK0
$$

$$ \det(A - \lambda I) = \det
LATEXBLOCK1
= \lambda^2 - 1 = 0 \implies \lambda_1 = 1, \lambda_2 = -1 $$

Find eigenvectors for each eigenvalue

Using the General Solution of ODE Systems knowledge point

$$ \lambda_1 = 1: \quad (A - I)\mathbf{v}_1 = \mathbf{0} \implies
LATEXBLOCK2

LATEXBLOCK3
=
LATEXBLOCK4
\implies \mathbf{v}_1 =
LATEXBLOCK5
$$

$$ \lambda_2 = -1: \quad (A + I)\mathbf{v}_2 = \mathbf{0} \implies
LATEXBLOCK6

LATEXBLOCK7
=
LATEXBLOCK8
\implies \mathbf{v}_2 =
LATEXBLOCK9
$$

$$ \mathbf{X}_c(t) = c_1 e^t
LATEXBLOCK10
+ c_2 e^{-t}
LATEXBLOCK11
$$

Find particular solution using undetermined coefficients

Assume a constant particular solution vector:

$$ \mathbf{X}_p =
LATEXBLOCK12
$$

$$ \mathbf{X}_p' = A\mathbf{X}_p + \mathbf{F}(t) \implies
LATEXBLOCK13
=
LATEXBLOCK14

LATEXBLOCK15
+
LATEXBLOCK16
$$

$$
LATEXBLOCK17
\implies a = -4, \ b = 5 \implies \mathbf{X}_p =
LATEXBLOCK18
$$

Combine complementary and particular solutions

Combine the solutions:

$$ \mathbf{X}(t) = \mathbf{X}_c(t) + \mathbf{X}_p = c_1 e^t
LATEXBLOCK19
+ c_2 e^{-t}
LATEXBLOCK20
+
LATEXBLOCK21
$$

Answer:

Use the method of undetermined coefficients to solve the given nonhomogeneous system. (Assume \(\mathbf{X} =

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

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

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

$$\begin{pmatrix} 3 \\ -1 \end{pmatrix}$$

+ c_2 e^{-t}

$$\begin{pmatrix} 1 \\ -1 \end{pmatrix}$$

+

$$\begin{pmatrix} -4 \\ 5 \end{pmatrix}$$

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