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

Show that \(x = 2e^{2t} + e^{-3t}\) 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 \(\frac{d^2x}{dt^2} + 5\frac{dx}{dt} + 4x = 0\), with x(0) = 5 and x′(0) = −8.

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 \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 0.75x = 0\), with x(0) = 3 and x′(0) = −1. Its exact solution is \(x = 3.5e^{-0.5t} - 0.5e^{-1.5t}\).

1. (a) Use Euler's method to estimate x(4) with step length 1 and with step length 0.5.
2. (b) Find the exact value of x(4).
3. (c) Find the percentage error of each estimate.

The displacement x cm of the end of a clamped ruler, t seconds after it is flicked, ignoring friction at the clamp, satisfies \(\frac{d^2x}{dt^2} + 6.25x = 0\), with x(0) = 0 and x′(0) = 2.5.

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 vertical displacement x mm of a washing machine casing satisfies \(\frac{d^2x}{dt^2} + 0.6\frac{dx}{dt} + 8x = 3\cos(\omega t)\). 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 = 1.5.
2. (b) Find the maximum displacement when ω = 2.8.
3. (c) Which driving frequency gives the larger motion? The natural frequency is √8 ≈ 2.83.

The concentration n mg per litre of a drug in a patient's blood, t hours after a dose, satisfies \(\frac{d^2n}{dt^2} + 0.6\frac{dn}{dt} + 5n = 4\), with n(0) = 1.5 and n′(0) = −8.

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 copper on a metals market is modelled by \(\frac{d^2p}{dt^2} + \frac{dp}{dt} + 4p = 14 - 3e^{-0.4t}\), where p is the price in dollars and t is the time in years. The price is now 5 dollars and is falling at 2 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 tonne
4. (d) the long-term price
5. (e) Suggest why the model is unlikely to be accurate.

Consider \(\frac{d^2x}{dt^2} + \left(\frac{dx}{dt}\right)^2 + x = 4 + 18t^2\), 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 \(x = 2t^2\) satisfies the equation and the starting values.
3. (c) Hence find the percentage error in each estimate in part (a).