Mixed practice: stability of second order equations Problem set B Worked solutions

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

1. (a) \(\frac{d^2x}{dt^2} + 11\frac{dx}{dt} + 18x = 0\)

   • Let y = dx/dt
   • \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -11y - 18x\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ -18 & -11 \end{pmatrix}\)

2. (b) \(\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 0\)

   • Let y = dx/dt
   • \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 6y - 25x\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ -25 & 6 \end{pmatrix}\)

3. (c) \(\frac{d^2x}{dt^2} = 8x - 5\frac{dx}{dt}\)

   • Replace dx/dt by y and d²x/dt² by dy/dt
   • \(\frac{dy}{dt} = 8x - 5y\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ 8 & -5 \end{pmatrix}\)

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

1. (a) \(\frac{d^2x}{dt^2} + 10\frac{dx}{dt} + 9x = 0\)

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

2. (b) \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} - 15x = 0\)

   • Characteristic equation \(\lambda^2 + 2\lambda - 15 = 0\)
   • λ = 3 and −5
   • Unstable: one eigenvalue is positive (a saddle)

3. (c) \(\frac{d^2x}{dt^2} - 8\frac{dx}{dt} + 15x = 0\)

   • Characteristic equation \(\lambda^2 - 8\lambda + 15 = 0\)
   • λ = 5 and 3
   • Unstable: both eigenvalues are positive

4. (d) \(\frac{d^2x}{dt^2} - 64x = 0\)

   • Characteristic equation \(\lambda^2 - 64 = 0\)
   • λ = 8 and −8
   • 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) \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 34x = 0\)

   • Characteristic equation \(\lambda^2 + 6\lambda + 34 = 0\)
   • λ = −3 ± 5i
   • Stable: the real part −3 is negative

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

   • Characteristic equation \(\lambda^2 - 4\lambda + 20 = 0\)
   • λ = 2 ± 4i
   • Unstable: the real part 2 is positive

3. (c) \(\frac{d^2x}{dt^2} + 25x = 0\)

   • Characteristic equation \(\lambda^2 + 25 = 0\)
   • λ = ±5i
   • Not stable: the real part is 0: x keeps oscillating and does not tend to 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.

   • \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -4y - 49x\)
   • Matrix \(\begin{pmatrix} 0 & 1 \\ -49 & -4 \end{pmatrix}\)

2. (b) Find the eigenvalues for c = 4 and say whether the motion dies away.

   • Characteristic equation \(\lambda^2 + 4\lambda + 49 = 0\)
   • λ = −2 ± 6.71i
   • Stable: the real part −2 is negative

3. (c) Find the eigenvalues for c = 20 and say whether the motion dies away.

   • Characteristic equation \(\lambda^2 + 20\lambda + 49 = 0\)
   • λ = −2.86 and −17.1
   • Stable: both eigenvalues are negative

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.

   • Matrix \(\begin{pmatrix} 0 & 1 \\ -7 & 0.8 \end{pmatrix}\)
   • Characteristic equation \(\lambda^2 - 0.8\lambda + 7 = 0\)
   • λ = 0.4 ± 2.62i

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

   • The real part is 0.4, which is positive
   • This means that the population swings above and below the target by larger and larger amounts, so it does not settle

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

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

   • Characteristic equation \(\lambda^2 + 2\lambda + 16 = 0\)
   • λ = −1 ± 3.87i
   • Stable: the real part −1 is negative

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

   • Characteristic equation \(\lambda^2 + 10\lambda + 16 = 0\)
   • λ = −2 and −8
   • 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 + 16 = 0, so λ = ±4i, 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 a for which the solution of \(\frac{d^2x}{dt^2} + a\frac{dx}{dt} + 5x = 0\) is stable. Give a reason.

• Characteristic equation λ² + aλ + 5 = 0: the sum of the roots is −a and the product is 5
• If the roots are complex, the real part is −a/2: stable if a > 0, unstable if a < 0, and not stable if a = 0
• If the roots are real, the product 5 is positive so they have the same sign, and the sum −a decides: both negative if a > 0
• So stable exactly when a > 0