Euler's method for a coupled system Problem set A

The system \(\dot x=3x - y,\ \dot y=x + 2y\) has \(x=2,\ y=1\) 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 \(\dot x=x - 2y,\ \dot y=3x\) has \(x=2,\ y=1\) 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 concentrations of two chemicals in a mixture from their equilibrium values are x and y (in mol/L) at time t minutes, and \(\dot x\) and \(\dot y\) are modelled by \(\dot x=-3y,\ \dot y=2x - y\). Initially \(x=3\) and \(y=1\).

1. (a) Write down the Euler recurrence relations with step length 0.1.
2. (b) Use your calculator to sketch x against t for \(0\le t\le10\). Label the first t-intercept and the first minimum point.
3. (c) Describe how x changes with t, and what this says about the concentrations in the long term.

The deviations, in hundreds, of the numbers of two competing species from their usual levels are x and y at time t years, modelled by \(\dot x=2y,\ \dot y=-3x + y\). Initially \(x=2\) 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 populations in words.

A student uses Euler's method with step length 0.1 on \(\dot x=-2x - 5y,\ \dot y=2x\) with \(x=4,\ y=1\) at \(t=0\). Every fifth row of the output is shown, to 3 significant figures: \(t=0:\ x=4.00,\ y=1.00\); \(t=0.5:\ x=-2.42,\ y=2.27\); \(t=1:\ x=-2.22,\ y=-0.723\); \(t=1.5:\ x=1.56,\ y=-1.30\); \(t=2:\ x=1.22,\ y=0.503\); \(t=2.5:\ x=-0.992,\ y=0.739\); \(t=3:\ x=-0.663,\ y=-0.340\).

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 \(\dot x=y,\ \dot y=-x\) 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 \(0\le t\le20\). What do you see?
3. (c) The exact solution and the Euler output do not agree. Why?