Coupled first order equationsAQA A-Level Further Maths: Mind map

Exam questions on Coupled first order equations

1. 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
2. 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
3. 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