Here we have a system of differential equations.  (dx/dt) = 11x + 8y,  (dy/dt) = 8x - y.  The coefficient matrix,  [11   8] [8   -1]  has eigenvectors:  [1]   and   [2] [-2]           [1]  With corresponding eigenvalues of -5 and 15.  what is the general solution of this system?

Express the given system of higher-order differential equations as a matrix system in normal form. x" + 5x+6y=0 y" - 5x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? O A. x₁ = x¹, x₂ =y' O B. X₁ =X, X₂=X', X3 = Y, X4=y' OC. x₁ = x, X₂=x", X3 = Y. X4=y" OD. x₁ = x¹, x₂ = x¹, x₂ =y', X4=y" Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) O A. O B. X₁ X₂ x₁² X1 x2 x3 X4 X1

A system of ordinary differential equations has 2 x 2 state matrix with eigenvalues and corresponding eigenvectors (eigenpairs) A1 = 5, v1 = and A2 = -6, Based on the given eigenpairs write down the general solution to the associated system of differential equations. The solution must be written as a |¤1(t) single vector like a =

Express the given system of higher-order differential equations as a matrix system in normal form. x" + 6x + 7y=0 y" - 2x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. X₁ = x¹, x₂ = y' O B. X₁ =X, X2₂=X'', X3 = Y, X4 =y" O c. x₁ = x¹, x₂ = x¹, X3 = y', X4 =y" O D. x₁ = x, X₂ = x', X3 = Y, X4 = y' Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) OA. B. X₁ X₂ X₁ S w X4 || X₁ X2 x1 3 ... W X4 ро

Let A be a 2 x 2 matrix with eigenvalues X₁ = -3 and >₂ = -1 and [1] an H Let x(t) be the position of a particle at time t. Suppose we solved the initial corresponding eigenvectors V₁ = value problem x = Ax, x(0) = [3] -ce + c₂e-t x(t) = = [2(t)] = [ qe * +ge + -3t Search and C2= -3t and V2 = to obtain then C1=

Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,

find the general solution of x‘= ax,exponentials of matrices must be explicitly calculated if used in your answer.