Question

Find the solution of the system x' = -4y, y' = -6x

where primes indicate derivatives with respect to t, that satisfies the initial condition x(0-2, y(0) =-2. y = ___
x = ___
Based on the general solution from which you obtained your particular solution, complete the following two statements: The critical point (0,0) is A. stable B. asymptotically stable C. unstable and is a A. center B. spiral C. node D. saddle point

Answer 1

Answer: The system of differential equations can be rewritten as a matrix equation:

[x', y']^T = [-4y, -6x]^T

The characteristic equation is obtained by setting the determinant of the matrix [A - λI] equal to 0, where A is the coefficient matrix, λ is an eigenvalue, and I is the identity matrix:

det([A - λI]) = det([-λ -4y; -6x, -λ]) = λ^2 + 24 = 0

Thus, λ = ±2i√6. The general solution to the system is then given by:

x(t) = c1cos(2√6t) + c2sin(2√6t)

y(t) = -2c1sin(2√6t) + 2c2cos(2√6t)

where c1 and c2 are constants determined by the initial conditions. Using the initial condition x(0) = -2, y(0) = -2, we can solve for c1 and c2:

-2 = c1cos(0) + c2sin(0)

-2 = -2c1sin(0) + 2c2cos(0)

c1 = -2

c2 = -2

So the particular solution to the system is:

x(t) = -2cos(2√6t) - 2sin(2√6t)

y(t) = 4sin(2√6t) - 4cos(2√6t)

The critical point (0,0) is a center, as the solution spirals towards the origin as t increases.

Explanation: