5.17 Phase portraitsIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

How do you write \frac{dx}{dt}=ax+by, \frac{dy}{dt}=cx+dy in matrix form?
What is a phase portrait?
How do you find the eigenvalues of M?
How do you find an eigenvector?
What is the general solution for real distinct eigenvalues?
Both eigenvalues real and positive?
Both eigenvalues real and negative?
Real eigenvalues of opposite signs?
Complex eigenvalues p\pm qi with p0?
Complex eigenvalues p\pm qi with p<0?
Imaginary eigenvalues \pm qi?
What happens if you start on an eigenvector?
How can you find whether a spiral is clockwise or anticlockwise?

Exam questions on 5.17 Phase portraits

The populations $x$ and $y$ (in hundreds) of two species of algae in a pond, $t$ weeks after they are introduced, satisfy $\frac{dx}{dt}=4x+y$ and $\frac{dy}{dt}=2x+3y$ . A GDC may be used.
Find an eigenvector of the matrix corresponding to the eigenvalue $2$ .2 marks
The deviations $x$ and $y$ (in thousands) of the populations of two interacting species from their equilibrium values satisfy $\frac{dx}{dt}=x+2y$ and $\frac{dy}{dt}=3x$ . A GDC may be used.
Find the equations of the two straight-line trajectories that pass through the origin.2 marks
The deviations $x$ and $y$ (in hundreds) of two interacting populations from their equilibrium values, $t$ years after a survey begins, satisfy $\frac{dx}{dt}=-x-4y$ and $\frac{dy}{dt}=x-y$ .
Find the eigenvalues of the matrix $\begin{pmatrix} -1 & -4 \\ 1 & -1 \end{pmatrix}$ that represents the system.3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).