Using matrix algebra, find a general solution to the following system of equations x' = 3x - 4y and y' = 4x - 7y

Question

Using matrix algebra, find a general solution to the following system of equations:

x' = 3x - 4y

y' = 4x - 7y

The general solution functions are: ( use c1 and c2 as the constants and enter the elements of the eigenvectors as the lowest integer values. If one element of an eigenvector has a negative value enter the first element as negative.)

x =

y =

Answer 1

Matrix A = [(3, - 4) (4, - 7)] Igen values: det[[(3 - λ, - 4) (4, - 7- λ)] = 0; λ² + 4λ - 5 = 0; λ₁ = 1; λ₂ = - 5.

Igen vector for λ₁ = 1: [(2, - 4) (4, - 8)][a b] = [0 0] 2a - 4b = 0; 4a - 8b = 0; a = 2 , b = 1 r = [2 1]

Igen vector for λ₂ = - 5: [8, - 4) (4, - 2)][a b] = [0, 0] 8a - 4b = 0; 4a - 2b = 0; a = 1 , b = 2 r = [1 2]

x(t) = 2c₁e^t + c₂e^-5t;

y(t) = c₁e^t + 2c₂e^-5t

This system of 2 ODE with 2 unknown functions x(t) and y(t).

I hope that we solved it right. Good luck!

Regards,

Yefim