Mixed practice: stability of second order equations Problem set A

Write each equation as a system of coupled first order equations, using y = dx/dt, and give the matrix of each system.

1. (a) \(\frac{d^2x}{dt^2} + 9\frac{dx}{dt} + 14x = 0\)
2. (b) \(\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + 10x = 0\)
3. (c) \(\frac{d^2x}{dt^2} = 7x - 3\frac{dx}{dt}\)

Decide whether the solution of each equation is stable. Give the eigenvalues.

1. (a) \(\frac{d^2x}{dt^2} + 8\frac{dx}{dt} + 7x = 0\)
2. (b) \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} - 10x = 0\)
3. (c) \(\frac{d^2x}{dt^2} - 7\frac{dx}{dt} + 12x = 0\)
4. (d) \(\frac{d^2x}{dt^2} - 49x = 0\)

Write each equation as a system and decide whether the solution is stable. Give the eigenvalues.

1. (a) \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 20x = 0\)
2. (b) \(\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + 5x = 0\)
3. (c) \(\frac{d^2x}{dt^2} + 16x = 0\)

The vertical displacement x metres of a car body on its suspension, t seconds after a bump, satisfies \(\frac{d^2x}{dt^2} + c\frac{dx}{dt} + 36x = 0\), where c is a positive constant.

1. (a) For c = 3, write the equation as a system and give its matrix.
2. (b) Find the eigenvalues for c = 3 and say whether the motion dies away.
3. (c) Find the eigenvalues for c = 15 and say whether the motion dies away.

The temperature x °C above target in a faulty heating system, t minutes after it is switched on, satisfies \(\frac{d^2x}{dt^2} - 0.6\frac{dx}{dt} + 5x = 0\).

1. (a) Find the eigenvalues of the system.
2. (b) Is the solution stable? Say what this means for the temperature.

Consider \(\frac{d^2x}{dt^2} + k\frac{dx}{dt} + 9x = 0\).

1. (a) Find the eigenvalues, and decide stability, when k = 2.
2. (b) Find the eigenvalues, and decide stability, when k = 10.
3. (c) Without solving, say what happens to the stability when k = 0 and when k is negative. Give a reason.

Find all values of b for which the solution of \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + bx = 0\) is stable. Give a reason.