Mixed practice: second order equations, exact solutions and models Problem set A

Show that \(x = e^{3t} + 2e^{-2t}\) satisfies \(\frac{d^2x}{dt^2} - \frac{dx}{dt} - 6x = 0\) and the starting values x(0) = 3 and x′(0) = −1.

A function satisfies \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} - 4x = 0\), with x(0) = 3 and x′(0) = −2.

1. (a) Write the equation as a system and give its matrix.
2. (b) Find the eigenvalues and an eigenvector for each.
3. (c) Find the exact solution for x in terms of t.
4. (d) Find x(1).

A function satisfies \(\frac{d^2x}{dt^2} + 1.5\frac{dx}{dt} + 0.5x = 0\), with x(0) = 2 and x′(0) = 0. Its exact solution is \(x = 4e^{-0.5t} - 2e^{-t}\).

1. (a) Use Euler's method to estimate x(3) with step length 1 and with step length 0.5.
2. (b) Find the exact value of x(3).
3. (c) Find the percentage error of each estimate.

The displacement x cm of the tip of a tuning fork, t seconds after it is struck, ignoring air resistance, satisfies \(\frac{d^2x}{dt^2} + 9x = 0\), with x(0) = 0 and x′(0) = 3.

1. (a) Use Euler's method with step length 0.1 to sketch the motion for 0 ≤ t ≤ 4.
2. (b) Estimate the amplitude, using the first peak, and the period, using the first two peaks. Give each to 2 significant figures.

The displacement x cm of a loudspeaker cone satisfies \(\frac{d^2x}{dt^2} + 0.4\frac{dx}{dt} + 5x = 2\sin(\omega t)\). It starts at rest at x = 0. Use Euler's method with step length 0.1 and technology to find the maximum displacement in the first 8 seconds.

1. (a) Find the maximum displacement when ω = 1.
2. (b) Find the maximum displacement when ω = 2.2.
3. (c) Which driving frequency gives the larger motion? The natural frequency is √5 ≈ 2.24.

The number n thousand of bacteria in a dish, t hours after an antibiotic is added, satisfies \(\frac{d^2n}{dt^2} + 1.2\frac{dn}{dt} + 9n = 6\), with n(0) = 2 and n′(0) = −12.

1. (a) Use Euler's method with step length 0.1 to describe the behaviour of n for 0 ≤ t ≤ 7.
2. (b) Explain why the model is unsatisfactory.

The price of shares in an engineering company is modelled by \(\frac{d^2p}{dt^2} + 0.8\frac{dp}{dt} + 3p = 12 - 2e^{-0.5t}\), where p is the price in dollars and t is the time in years. The price is now 6 dollars and is falling at 1.8 dollars per year. Use Euler's method with step length 0.1 to find, to 1 decimal place, according to the model:

1. (a) when to buy
2. (b) how long to hold
3. (c) the profit per share
4. (d) the long-term price
5. (e) Suggest why the model is unlikely to be accurate.

Consider \(\frac{d^2x}{dt^2} + \left(\frac{dx}{dt}\right)^2 + x = 2 + 5t^2\), with x(0) = 0 and x′(0) = 0.

1. (a) Use Euler's method with step length 0.1 to estimate x(1) and x(2).
2. (b) Show that \(x = t^2\) satisfies the equation and the starting values.
3. (c) Hence find the percentage error in each estimate in part (a).