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

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 \(t_n\).

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

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

   • \(x_1=2+0.1[3(2)+(-1)(1)]=2.5\)
   • \(y_1=1+0.1[1(2)+(2)(1)]=1.4\)

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

   • \(x_2=2.5+0.1[3(2.5)+(-1)(1.4)]=3.11\)
   • \(y_2=1.4+0.1[1(2.5)+(2)(1.4)]=1.93\)
   • Use \(x_1\) and \(y_1\) in both

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.

   • \(x_{n+1}=x_n+0.1(x_n - 2y_n),\ y_{n+1}=y_n+0.1(3x_n)\)
   • From the calculator: scatter plot of x against t. x changes sign between \(t=0.6\) and \(t=0.7\)
   • First minimum about \((1.4,\ -5.65)\)
   • 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 \((2,1)\), anticlockwise
   • It crosses the y-axis at about \(y=3.8\) and the negative x-axis at about \(x=-4.9\)

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.

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

2. (b) Use your calculator to sketch x against t for \(0\le t\le10\). Label the first t-intercept and the first minimum point.

   • From the calculator: scatter plot of x against t
   • x changes sign between \(t=0.5\) and \(t=0.6\)
   • First minimum about \((1.2,\ -2.4)\)

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

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

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 \(0\le t\le4\).

   • \(x_{n+1}=x_n+0.1(2y_n),\ y_{n+1}=y_n+0.1(-3x_n + y_n)\)
   • From the calculator: scatter plot of \(y_n\) against \(x_n\)
   • A spiral out, clockwise, crossing the y-axis at about \(y=-3.8\) and the negative x-axis at about \(x=-5.65\)

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

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

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.

   • 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 \(10\): \(\lambda^2+2\lambda+10=0\)
   • \(\lambda=-1\pm3i\)
   • 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.

   • \(q=3\), so \(\pi\div q=1.05\)
   • The two sign changes in the table are in intervals about 1.0 apart, so the estimates agree

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?

   • Trace 0, determinant \(1\): \(\lambda^2+1=0\)
   • \(\lambda=\pm i\): 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 \(0\le t\le20\). What do you see?

   • \(x_{n+1}=x_n+0.1(y_n),\ y_{n+1}=y_n+0.1(-x_n)\)
   • 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 \(2.70\) at \(t=20\)

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