Coupled first-order differential equationsEdexcel A-Level Further Maths: Mind map

Exam questions on Coupled first-order differential equations

1. 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
2. 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
3. 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