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4. (5 pts total) Consider the homogeneous system of equations frac{d}{dt} egin{bmatrix} x(t) \ y(t) end{bmatrix} = egin{bmatrix} 2 & 3 \ 3 & 2 end{bmatrix} egin{bmatrix} x(t) \ y(t) end{bmatrix}. (2) a. Check (find derivatives and show LHS=RHS) that vec{x}_1(t) = egin{bmatrix} 4e^{-t} \ -4e^{-t} end{bmatrix} is a solution of the above equation (2). b. Two independent eigenvectors of the matrix egin{bmatrix} 2 & 3 \ 3 & 2 end{bmatrix} are: vec{u}_1 = egin{bmatrix} 1 \ -1 end{bmatrix}, with eigenvalue lambda = r_1, and vec{u}_2 = egin{bmatrix} 1 \ 1 end{bmatrix}, with eigenvalue lambda = r_2. Use the definition of eigenvalue: Avec{u} = lambda vec{u} to find the corresponding eigenvalues r_1 and r_2. (fill in each blank with one real number, in the correct order — they are not interchangeable.) r_1 = ______ r_2 = ______ c. The general solution set of the above system (2) of differential equations is a 2-dimensional set. Using the fact that the vector-valued function vec{x}(t) = e^{lambda t} vec{u} is a solution of (2) for each eigenvalue lambda and eigenvector vec{u}, fill in the blanks with column vectors to state the solution set: With scalar coefficients C_1, C_2 in mathbb{R}, solution set = {C_1______ + C_2______}. d. For your answer in part c., which scalars give the solution vec{x}_1(t) from part a.? C_1 = C_2 =

          4. (5 pts total) Consider the homogeneous system of equations
frac{d}{dt} egin{bmatrix} x(t) \ y(t) end{bmatrix} = egin{bmatrix} 2 & 3 \ 3 & 2 end{bmatrix} egin{bmatrix} x(t) \ y(t) end{bmatrix}. (2)
a. Check (find derivatives and show LHS=RHS) that vec{x}_1(t) = egin{bmatrix} 4e^{-t} \ -4e^{-t} end{bmatrix} is a solution of the above equation (2).
b. Two independent eigenvectors of the matrix egin{bmatrix} 2 & 3 \ 3 & 2 end{bmatrix} are:
vec{u}_1 = egin{bmatrix} 1 \ -1 end{bmatrix}, with eigenvalue lambda = r_1, and
vec{u}_2 = egin{bmatrix} 1 \ 1 end{bmatrix}, with eigenvalue lambda = r_2.
Use the definition of eigenvalue: Avec{u} = lambda vec{u} to find the corresponding eigenvalues r_1 and r_2. (fill in each blank with one real number, in the correct order — they are not interchangeable.)
r_1 = ______ r_2 = ______
c. The general solution set of the above system (2) of differential equations is a 2-dimensional set. Using the fact that the vector-valued function vec{x}(t) = e^{lambda t} vec{u} is a solution of (2) for each eigenvalue lambda and eigenvector vec{u}, fill in the blanks with column vectors to state the solution set:
With scalar coefficients C_1, C_2 in mathbb{R}, solution set = {C_1______ + C_2______}.
d. For your answer in part c., which scalars give the solution vec{x}_1(t) from part a.?
C_1 =
C_2 =

$4. (5 pts total) Consider the homogeneous system of equations fracddt eginbmatrix x(t) y(t) endbmatrix = eginbmatrix 2 3 3 2 endbmatrix eginbmatrix x(t) y(t) endbmatrix. (2) a. Check (find derivatives and show LHS=RHS) that vecx1(t) = eginbmatrix 4e^-t -4e^-t endbmatrix is a solution of the above equation (2). b. Two independent eigenvectors of the matrix eginbmatrix 2 3 3 2 endbmatrix are: vecu1 = eginbmatrix 1 -1 endbmatrix, with eigenvalue lambda = r1, and vecu2 = eginbmatrix 1 1 endbmatrix, with eigenvalue lambda = r2. Use the definition of eigenvalue: Avecu = lambda vecu to find the corresponding eigenvalues r1 and r2. (fill in each blank with one real number, in the correct order — they are not interchangeable.) r1 = r2 = c. The general solution set of the above system (2) of differential equations is a 2-dimensional set. Using the fact that the vector-valued function vecx(t) = e^lambda t vecu is a solution of (2) for each eigenvalue lambda and eigenvector vecu, fill in the blanks with column vectors to state the solution set: With scalar coefficients C1, C2 in mathbbR, solution set = C1 + C2. d. For your answer in part c., which scalars give the solution vecx1(t) from part a.? C1 = C2 =$

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