Euler's method for a coupled system Problem set B

The system \(\dot x=2x - 3y,\ \dot y=2x + y\) has \(x=3,\ y=2\) when \(t=0\). Use Euler's method with a step length of 0.1.

1. (a) Write down the recurrence relations for \(x_{n+1}\), \(y_{n+1}\) and \(t_n\).
2. (b) Find \(x_1\) and \(y_1\).
3. (c) Find \(x_2\) and \(y_2\).

The system \(\dot x=x - y,\ \dot y=3x + y\) has \(x=1,\ y=0\) when \(t=0\). Use Euler's method with a step length of 0.1 on your calculator.

1. (a) Sketch a graph of x against t for \(0\le t\le5\). Label the first t-intercept and the first minimum point.
2. (b) Sketch the phase portrait of y against x for \(0\le t\le5\).

The deviations of the temperature (x, in °C) and the humidity (y, in %) of a greenhouse from their set values at time t hours are modelled by \(\dot x=-x - 3y,\ \dot y=2x - y\). Initially \(x=5\) and \(y=2\).

1. (a) Write down the Euler recurrence relations with step length 0.1.
2. (b) Use your calculator to sketch x against t for \(0\le t\le8\). Label the first t-intercept and the first minimum point.
3. (c) Describe how x changes with t, and what this says about the greenhouse in the long term.

The deviations of the prices (in euros) of two shares from their usual values are x and y at time t weeks, modelled by \(\dot x=3y,\ \dot y=-2x + y\). Initially \(x=1\) and \(y=0\).

1. (a) Use Euler's method with step length 0.1 to sketch the phase portrait of y against x for \(0\le t\le4\).
2. (b) Describe the long-term behaviour of the two prices in words.

A student uses Euler's method with step length 0.1 on \(\dot x=-x - 5y,\ \dot y=2x - y\) with \(x=3,\ y=2\) at \(t=0\). Every fifth row of the output is shown, to 3 significant figures: \(t=0:\ x=3.00,\ y=2.00\); \(t=0.5:\ x=-2.76,\ y=1.30\); \(t=1:\ x=-1.36,\ y=-1.49\); \(t=1.5:\ x=1.98,\ y=-0.533\); \(t=2:\ x=0.476,\ y=1.03\); \(t=2.5:\ x=-1.32,\ y=0.140\); \(t=3:\ x=-0.0499,\ y=-0.669\).

1. (a) Does x oscillate? Give evidence from the table.
2. (b) Find the eigenvalues of the system, to 3 significant figures. Does the sign of the real part agree with how the values of x behave?
3. (c) Use the imaginary part q to estimate the time between two successive t-intercepts of x, using \(\pi\div q\). Compare with the table.

A system has \(\dot x=2y,\ \dot y=-2x\) with \(x=1\) and \(y=0\) when \(t=0\).

1. (a) Find the eigenvalues. What shape does the exact solution make on the phase portrait?
2. (b) Use Euler's method with step length 0.1 to sketch the phase portrait of y against x for \(0\le t\le10\). What do you see?
3. (c) The exact solution and the Euler output do not agree. Why?