Mixed practice: stability of second order equations Problem set A One question at a time

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

1. (a) \(\frac{d^2x}{dt^2} + 9\frac{dx}{dt} + 14x = 0\)

   • Let y = dx/dt
   • \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -9y - 14x\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ -14 & -9 \end{pmatrix}\)

2. (b) \(\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + 10x = 0\)

   • Let y = dx/dt
   • \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 2y - 10x\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ -10 & 2 \end{pmatrix}\)

3. (c) \(\frac{d^2x}{dt^2} = 7x - 3\frac{dx}{dt}\)

   • Replace dx/dt by y and d²x/dt² by dy/dt
   • \(\frac{dy}{dt} = 7x - 3y\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ 7 & -3 \end{pmatrix}\)

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

1. (a) \(\frac{d^2x}{dt^2} + 8\frac{dx}{dt} + 7x = 0\)

   • Characteristic equation \(\lambda^2 + 8\lambda + 7 = 0\)
   • λ = −1 and −7
   • Stable: both eigenvalues are negative

2. (b) \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} - 10x = 0\)

   • Characteristic equation \(\lambda^2 + 3\lambda - 10 = 0\)
   • λ = 2 and −5
   • Unstable: one eigenvalue is positive (a saddle)

3. (c) \(\frac{d^2x}{dt^2} - 7\frac{dx}{dt} + 12x = 0\)

   • Characteristic equation \(\lambda^2 - 7\lambda + 12 = 0\)
   • λ = 4 and 3
   • Unstable: both eigenvalues are positive

4. (d) \(\frac{d^2x}{dt^2} - 49x = 0\)

   • Characteristic equation \(\lambda^2 - 49 = 0\)
   • λ = 7 and −7
   • Unstable: one eigenvalue is positive (a saddle)

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

1. (a) \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 20x = 0\)

   • Characteristic equation \(\lambda^2 + 4\lambda + 20 = 0\)
   • λ = −2 ± 4i
   • Stable: the real part −2 is negative

2. (b) \(\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + 5x = 0\)

   • Characteristic equation \(\lambda^2 - 2\lambda + 5 = 0\)
   • λ = 1 ± 2i
   • Unstable: the real part 1 is positive

3. (c) \(\frac{d^2x}{dt^2} + 16x = 0\)

   • Characteristic equation \(\lambda^2 + 16 = 0\)
   • λ = ±4i
   • Not stable: the real part is 0: x keeps oscillating and does not tend to 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.

   • \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -3y - 36x\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ -36 & -3 \end{pmatrix}\)

2. (b) Find the eigenvalues for c = 3 and say whether the motion dies away.

   • Characteristic equation \(\lambda^2 + 3\lambda + 36 = 0\)
   • λ = −1.5 ± 5.81i
   • Stable: the real part −1.5 is negative

3. (c) Find the eigenvalues for c = 15 and say whether the motion dies away.

   • Characteristic equation \(\lambda^2 + 15\lambda + 36 = 0\)
   • λ = −3 and −12
   • Stable: both eigenvalues are negative

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.

   • Matrix \(\begin{pmatrix} 0 & 1 \\ -5 & 0.6 \end{pmatrix}\)
   • Characteristic equation \(\lambda^2 - 0.6\lambda + 5 = 0\)
   • λ = 0.3 ± 2.22i

2. (b) Is the solution stable? Say what this means for the temperature.

   • The real part is 0.3, which is positive
   • This means that the temperature swings above and below the target by larger and larger amounts, so the system does not settle

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

1. (a) Find the eigenvalues, and decide stability, when k = 2.

   • Characteristic equation \(\lambda^2 + 2\lambda + 9 = 0\)
   • λ = −1 ± 2.83i
   • Stable: the real part −1 is negative

2. (b) Find the eigenvalues, and decide stability, when k = 10.

   • Characteristic equation \(\lambda^2 + 10\lambda + 9 = 0\)
   • λ = −1 and −9
   • Stable: both eigenvalues are negative

3. (c) Without solving, say what happens to the stability when k = 0 and when k is negative. Give a reason.

   • When k = 0: λ² + 9 = 0, so λ = ±3i, real part 0: x does not tend to 0
   • When k < 0: the sum of the eigenvalues is −k > 0, so the real part, or at least one real eigenvalue, is positive: unstable

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.

• Characteristic equation λ² + 4λ + b = 0, so λ = −2 ± √(4 − b)
• If b > 4 the roots are complex with real part −2: stable
• If 0 < b ≤ 4 the roots are real and √(4 − b) < 2, so both are negative: stable
• If b = 0 one root is 0, and if b < 0 one root is positive: not stable. So stable exactly when b > 0