Mixed practice: stability of second order equations Problem set B

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

1. (a) \(\frac{d^2x}{dt^2} + 11\frac{dx}{dt} + 18x = 0\)
2. (b) \(\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 0\)
3. (c) \(\frac{d^2x}{dt^2} = 8x - 5\frac{dx}{dt}\)

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

1. (a) \(\frac{d^2x}{dt^2} + 10\frac{dx}{dt} + 9x = 0\)
2. (b) \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} - 15x = 0\)
3. (c) \(\frac{d^2x}{dt^2} - 8\frac{dx}{dt} + 15x = 0\)
4. (d) \(\frac{d^2x}{dt^2} - 64x = 0\)

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

1. (a) \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 34x = 0\)
2. (b) \(\frac{d^2x}{dt^2} - 4\frac{dx}{dt} + 20x = 0\)
3. (c) \(\frac{d^2x}{dt^2} + 25x = 0\)

The angle x radians of a swing door from its closed position, t seconds after it is released, satisfies \(\frac{d^2x}{dt^2} + c\frac{dx}{dt} + 49x = 0\), where c is a positive constant.

1. (a) For c = 4, write the equation as a system and give its matrix.
2. (b) Find the eigenvalues for c = 4 and say whether the motion dies away.
3. (c) Find the eigenvalues for c = 20 and say whether the motion dies away.

The number x thousand of fish above the target level in a badly managed lake, t years after stocking, satisfies \(\frac{d^2x}{dt^2} - 0.8\frac{dx}{dt} + 7x = 0\).

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

Consider \(\frac{d^2x}{dt^2} + k\frac{dx}{dt} + 16x = 0\).

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 a for which the solution of \(\frac{d^2x}{dt^2} + a\frac{dx}{dt} + 5x = 0\) is stable. Give a reason.