Euler's method for a coupled system Problem set B One question at a time

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 \(t_n\).

   • \(x_{n+1}=x_n+0.1(2x_n - 3y_n)\)
   • \(y_{n+1}=y_n+0.1(2x_n + y_n)\)
   • \(t_n=0.1n\)

2. (b) Find \(x_1\) and \(y_1\).

   • \(x_1=3+0.1[2(3)+(-3)(2)]=3\)
   • \(y_1=2+0.1[2(3)+(1)(2)]=2.8\)

3. (c) Find \(x_2\) and \(y_2\).

   • \(x_2=3+0.1[2(3)+(-3)(2.8)]=2.76\)
   • \(y_2=2.8+0.1[2(3)+(1)(2.8)]=3.68\)
   • Use \(x_1\) and \(y_1\) in both

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.

   • \(x_{n+1}=x_n+0.1(x_n - y_n),\ y_{n+1}=y_n+0.1(3x_n + y_n)\)
   • From the calculator: scatter plot of x against t. x changes sign between \(t=1\) and \(t=1.1\)
   • First minimum about \((2.4,\ -10.9)\)
   • An oscillation with growing amplitude

2. (b) Sketch the phase portrait of y against x for \(0\le t\le5\).

   • From the calculator: scatter plot of \(y_n\) against \(x_n\)
   • The spiral goes out from \((1,0)\), anticlockwise
   • It crosses the y-axis at about \(y=5.1\) and the negative x-axis at about \(x=-8.7\)

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.

   • \(x_{n+1}=x_n+0.1(-x_n - 3y_n)\)
   • \(y_{n+1}=y_n+0.1(2x_n - y_n)\)
   • \(t_n=0.1n\)

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.

   • From the calculator: scatter plot of x against t
   • x changes sign between \(t=0.4\) and \(t=0.5\)
   • First minimum about \((0.9,\ -2.85)\)

3. (c) Describe how x changes with t, and what this says about the greenhouse in the long term.

   • An oscillation with decreasing amplitude, so x tends to 0 and the temperature returns to its set value

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 \(0\le t\le4\).

   • \(x_{n+1}=x_n+0.1(3y_n),\ y_{n+1}=y_n+0.1(-2x_n + y_n)\)
   • From the calculator: scatter plot of \(y_n\) against \(x_n\)
   • A spiral out, clockwise, crossing the y-axis at about \(y=-1.3\) and the negative x-axis at about \(x=-2.8\)

2. (b) Describe the long-term behaviour of the two prices in words.

   • The spiral goes out, so both deviations oscillate with growing size: the prices swing further from their usual values each time

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.

   • x goes positive, then negative, then positive again
   • x changes sign between 0 and 0.5, and again between 1 and 1.5: an oscillation

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?

   • Trace \(-2\), determinant \(11\): \(\lambda^2+2\lambda+11=0\)
   • \(\lambda=-1\pm3.16i\)
   • Negative real part: the swings shrink as t increases, and the table shows this

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.

   • \(q=3.16\), so \(\pi\div q=0.993\)
   • The two sign changes in the table are in intervals about 1.0 apart, so the estimates agree

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?

   • Trace 0, determinant \(4\): \(\lambda^2+4=0\)
   • \(\lambda=\pm2i\): the real part is 0, so the exact solution is a closed curve

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?

   • \(x_{n+1}=x_n+0.1(2y_n),\ y_{n+1}=y_n+0.1(-2x_n)\)
   • From the calculator: scatter plot of \(y_n\) against \(x_n\)
   • A spiral going out: the distance from the origin is \(1\) at the start and about \(7.11\) at \(t=10\)

3. (c) The exact solution and the Euler output do not agree. Why?

   • Euler's method moves along the tangent for each step, and the tangent to a closed curve points outside it
   • The error adds up at every step, so the points drift outwards