Coupled first-order differential equationsEdexcel A-Level Further Maths: Flashcards

What these 13 flashcards ask

What is a coupled system of first-order equations?
Form of the linear coupled system in the spec?
Main method to solve a coupled system?
From \dot{x}=2x+y, what is y?
Which equation do you use to find the second variable?
For \dot{x}=ax+by, \dot{y}=cx+dy, the elimination gives which equation for x?
How many arbitrary constants in the general solution of a coupled pair?
How do you find a PI for constant forcing?
General solution structure with forcing terms?
What does a -2y term in \dot{x} and +2x in \dot{y} suggest?
How do initial conditions work for a coupled system?
Give a limitation of a linear predator-prey model.
What happens to x and y in the long term if the exponents are negative and forcing is constant?

Exam questions on Coupled first-order differential equations

Two quantities $x$ and $y$ satisfy the pair of coupled differential equations $\frac{dx}{dt}=2x+y$ and $\frac{dy}{dt}=3x+4y$ .
Hence find $y$ in terms of $t$ .2 marks
In a model of two species, the populations $x$ and $y$ (in thousands) at time $t$ years satisfy $\frac{dx}{dt}=3x-2y$ and $\frac{dy}{dt}=2x-2y$ .
Find the general solution for $y$ .2 marks
The amounts $x$ and $y$ of a chemical in two connected vessels at time $t$ minutes satisfy $\frac{dx}{dt}=-2x+y$ and $\frac{dy}{dt}=x-2y+9$ , where the term $+9$ is a constant supply of the chemical into the second vessel.
Show that $\ddot{x}+4\dot{x}+3x=9$ .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).