(a) To convert the second order differential equation to a system of first order differential equations, we can introduce a new variable y(t) = dx/dt. Then, the original equation can be rewritten as:

dx/dt = y
dy/dt = -ay - bx

This is a system of first order differential equations. In matrix form, this system can be written as:

[dx/dt] = [0 1] [x]
[dy/dt] [-b -a] [y]

(b) For a = 1 and b = -2, the matrix associated with the system is:

[0 1]
[-2 -1]

The eigenvalues of this matrix are the roots of the characteristic equation, which is given by:

det(A - λI) = 0

where A is the matrix, I is the identity matrix, and λ are the eigenvalues. For this matrix, the characteristic equation is:

λ^2 + λ + 2 = 0

The roots of this equation are λ = -1/2 ± √7/2 i. These are the eigenvalues of the matrix.

The eigenvectors are the solutions to the equation (A - λI)v = 0, where v is the eigenvector. For each eigenvalue, we get:

For λ = -1/2 + √7/2 i:

[1/2 - √7/2 i 1] [v1] = [0]
[-2 -1] [v2] [0]

This gives the system of equations:

(1/2 - √7/2 i)v1 + v2 = 0
-2v1 - v2 = 0

Solving this system gives the eigenvector v = [1, 2 + √7 i].

Similarly, for λ = -1/2 - √7/2 i, we get the eigenvector v = [1, 2 - √7 i].

The general solution of the system is then given by:

x(t) = c1 * e^(-1/2 + √7/2 i)t * [1, 2 + √7 i] + c2 * e^(-1/2 - √7/2 i)t * [1, 2 - √7 i]

where c1 and c2 are constants determined by initial conditions.

(c) The equilibrium point of the system is the origin, since this is the point where dx/dt = dy/dt = 0. The type of the equilibrium point is determined by the eigenvalues of the matrix. Since the real part of both eigenvalues is negative, the origin is a stable spiral. The phase plane of the system will show trajectories spiraling into the origin.

Question

(a) To convert the second order differential equation to a system of first order differential equations, we can introduce a new variable y(t) = dx/dt. Then, the original equation can be rewritten as:

dx/dt = y
dy/dt = -ay - bx

This is a system of first order differential equations. In matrix form, this system can be written as:

[dx/dt]   = [0  1] [x]
[dy/dt]     [-b -a] [y]

(b) For a = 1 and b = -2, the matrix associated with the system is:

[0  1]
[-2 -1]

The eigenvalues of this matrix are the roots of the characteristic equation, which is given by:

det(A - λI) = 0

where A is the matrix, I is the identity matrix, and λ are the eigenvalues. For this matrix, the characteristic equation is:

λ^2 + λ + 2 = 0

The roots of this equation are λ = -1/2 ± √7/2 i. These are the eigenvalues of the matrix.

The eigenvectors are the solutions to the equation (A - λI)v = 0, where v is the eigenvector. For each eigenvalue, we get:

For λ = -1/2 + √7/2 i:

[1/2 - √7/2 i  1] [v1] = [0]
[-2 -1] [v2]   [0]

This gives the system of equations:

(1/2 - √7/2 i)v1 + v2 = 0
-2v1 - v2 = 0

Solving this system gives the eigenvector v = [1, 2 + √7 i].

Similarly, for λ = -1/2 - √7/2 i, we get the eigenvector v = [1, 2 - √7 i].

The general solution of the system is then given by:

x(t) = c1 * e^(-1/2 + √7/2 i)t * [1, 2 + √7 i] + c2 * e^(-1/2 - √7/2 i)t * [1, 2 - √7 i]

where c1 and c2 are constants determined by initial conditions.

(c) The equilibrium point of the system is the origin, since this is the point where dx/dt = dy/dt = 0. The type of the equilibrium point is determined by the eigenvalues of the matrix. Since the real part of both eigenvalues is negative, the origin is a stable spiral. The phase plane of the system will show trajectories spiraling into the origin.

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