3. Consider the following system of equations, x(t) = (x, y), - x' = (3-1)x 5y y' = 2x+(3+ 1)y (a) Determine the eigenvalues of the system in terms of ẞ. (b) Determine the eigenvectors. (c) Find the general solution to the system in terms of ẞ. (d) Describe the different possible behaviours of solutions X (t) = (x, y) in the phase plane, for different values of the parameter B.

please solve it on paper

Consider the linear system 3 *' = 27 a. Find the eigenvalues and eigenvectors for the coefficient matrix. help (numbers) help (matrices) A1 = , 01 = and help (numbers) help (matrices) 12 = , v2 = Find the real-valued solution to the initial value problem x = -3x1 – 2x2, x = 5x1+3x2, x1(0) = -4, x2(0) = 10. Use t as the independent variable in your answers. help (formulas) T2(t) help (formulas) OK Learn more Cookies help us deliver our services. By using our services, you agree to our use of cookies.

3. Consider the system of equations given by: a': -2 I. -3 a. Find a fundamental matrix for the system. x (t) = b. Find the matrix exponential, ¥(t) = e4", of the system. V(t) = (:) c. Solve the initial value problem with æ(0) = (. using the matrix exponential found in Part b. x(t) =

Find the eigenvalues and eigenvectors for the coefficient matrix.

Use (a) to solve the system x = (: :) . Х. -8 7 NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly independent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. a) is as follows Show from first principles, i.e., by using the definition of linear independence, that if x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = et (u cos bt – w sin bt) and Z(t) = eat (u sin bt + w cos bt) are linearly independent solutions of X = AX.

Find the solutions of the system of linear equations if the eigenvalues of the matrix A are

please solve it on paper

For the linear system y' = 3 Find the eigenvalues and eigenvectors for the coefficient matrix. A1 = , and A2 = v2 =

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