Coupled first order equationsAQA A-Level Further Maths: Flashcards

What these 13 flashcards ask

What is a coupled first order system?
How do you solve a coupled pair?
Second order equation from x'=ax+by, y'=cx+dy?
How do you find y once x is known?
What does a term -by in \frac{dx}{dt} mean in a predator-prey model with x prey?
What does +cx in \frac{dy}{dt} mean for predators?
What does -dy in \frac{dy}{dt} mean?
What does an equilibrium mean?
What do complex roots of the auxiliary equation mean?
What do negative real parts of the roots mean?
How do predator and prey cycles relate?
x'=3x+2y, y'=x+2y: second order equation for x?
How do you use x(0) and y(0)?

Exam questions on Coupled first order equations

The functions $x$ and $y$ of $t$ satisfy $\frac{dx}{dt}=3x+2y$ and $\frac{dy}{dt}=x+2y$ .
Given that $x=Ae^{t}+Be^{4t}$ , find $y$ in terms of $t$ , $A$ and $B$ .2 marks
In a model of a predator-prey system, $x$ and $y$ are the sizes (in hundreds) of the prey and predator populations at time $t$ years, where $\frac{dx}{dt}=2x-3y$ and $\frac{dy}{dt}=x-2y$ .
Show that $x''-x=0$ .2 marks
Salt is exchanged between two connected tanks, $P$ and $Q$ . The masses $x$ kg and $y$ kg of salt in $P$ and $Q$ at time $t$ minutes satisfy $\frac{dx}{dt}=-2x+y$ and $\frac{dy}{dt}=2x-3y$ .
Show that $x''+5x'+4x=0$ .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).