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(a) (i) Express the following...

(a) (i) Express the following inhomogeneous system of first-order

differential equations for x(t) and y(t) in matrix form.

.

x = −x + 7y + cos t − 8 sin t

.

y = 7x−y−6 cos t + sin t [1]

(ii) Write down, also in matrix form, the corresponding homogeneous

system of equations.

[1]

(iii) Find the eigenvalues of the matrix of coefficients and an

eigenvector corresponding to each eigenvalue. Hence write down

the complementary function for the system of equations. [6]

(iv) A particular integral of the inhomogeneous system of equations is

of the form

[x]= [a sin t + b cos t]

[y]= [c sin t + d cos t]

where a, b, c and d are constants to be determined. By

substituting in the original inhomogeneous system of equations,

find a system of four linear equations satisfied by a, b, c and d.

Check (by substitution) that the solution to these equations is

given by

a = 0, b= 1, c= 1 and d = 0.

Hence determine the particular integral to the inhomogeneous

system. [4]

(v) State the general solution to the original inhomogeneous system

of differential equations. [1]

(vi) Find the solution that satisfies the initial conditions x(0) = 1 and

y(0) = −2. [2]

(b) An object moves in the plane in such a way that its Cartesian

coordinates (x, y) at time t satisfy the following homogeneous system

of second-order differential equations.

.

x = −x + 7y

.

y = 7x−y

Express the system in matrix form.

Which eigenvalue of the matrix of coefficients leads to solutions of the

system that describe simple harmonic motion along a straight line?

Briefly justify your answer.

Determine the angular frequency of this motion and a vector giving

the direction of the simple harmonic motion. [4]

differential equations for x(t) and y(t) in matrix form.

.

x = −x + 7y + cos t − 8 sin t

.

y = 7x−y−6 cos t + sin t [1]

(ii) Write down, also in matrix form, the corresponding homogeneous

system of equations.

[1]

(iii) Find the eigenvalues of the matrix of coefficients and an

eigenvector corresponding to each eigenvalue. Hence write down

the complementary function for the system of equations. [6]

(iv) A particular integral of the inhomogeneous system of equations is

of the form

[x]= [a sin t + b cos t]

[y]= [c sin t + d cos t]

where a, b, c and d are constants to be determined. By

substituting in the original inhomogeneous system of equations,

find a system of four linear equations satisfied by a, b, c and d.

Check (by substitution) that the solution to these equations is

given by

a = 0, b= 1, c= 1 and d = 0.

Hence determine the particular integral to the inhomogeneous

system. [4]

(v) State the general solution to the original inhomogeneous system

of differential equations. [1]

(vi) Find the solution that satisfies the initial conditions x(0) = 1 and

y(0) = −2. [2]

(b) An object moves in the plane in such a way that its Cartesian

coordinates (x, y) at time t satisfy the following homogeneous system

of second-order differential equations.

.

x = −x + 7y

.

y = 7x−y

Express the system in matrix form.

Which eigenvalue of the matrix of coefficients leads to solutions of the

system that describe simple harmonic motion along a straight line?

Briefly justify your answer.

Determine the angular frequency of this motion and a vector giving

the direction of the simple harmonic motion. [4]

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