Coupled first-order differential equationsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Two quantities $x$ and $y$ satisfy the pair of coupled differential equations $\frac{dx}{dt}=2x+y$ and $\frac{dy}{dt}=3x+4y$.

   (a)

   Eliminating $y$ gives which second-order differential equation for $x$?

   [1 mark]

   • A$\ddot{x}+6\dot{x}+5x=0$
   • B$\ddot{x}-6\dot{x}+11x=0$
   • C$\ddot{x}-6\dot{x}+5x=0$
   • D$\ddot{x}-6\dot{x}+8x=0$

   (b)

   Find the general solution for $x$.

   [1 mark]

   • A$x=Ae^{t}+Be^{5t}$
   • B$x=Ae^{-t}+Be^{-5t}$
   • C$x=(A+Bt)e^{3t}$
   • D$x=Ae^{t}+Be^{6t}$

   (c)

   Hence find $y$ in terms of $t$.

   [2 marks]

   Total for question 1: 4 marks

2. 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 $\frac{dy}{dt}=2x-2y$.

   (a)

   Which description of the two species fits the model?

   [1 mark]

   • ABoth are prey, competing for the same food.
   • B$x$ is the prey and $y$ the predator: $x$ would grow without $y$, while $y$ would decline without $x$.
   • C$x$ is the predator and $y$ the prey: $x$ would decline without $y$, while $y$ would grow without $x$.
   • DThe species do not interact.

   (b)

   Eliminating $x$ gives which second-order differential equation for $y$?

   [1 mark]

   • A$\ddot{y}+\dot{y}-2y=0$
   • B$\ddot{y}-\dot{y}+2y=0$
   • C$\ddot{y}+\dot{y}+2y=0$
   • D$\ddot{y}-\dot{y}-2y=0$

   (c)

   Find the general solution for $y$.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Show that $\ddot{x}+4\dot{x}+3x=9$.

   [3 marks]

   (b)

   Given that $x=0$ and $y=0$ when $t=0$, find $x$ and $y$ in terms of $t$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a model, $x$ and $y$ (in hundreds) are the numbers of prey and predators in a region at time $t$ years, where $\frac{dx}{dt}=3x-2y$ and $\frac{dy}{dt}=x$. Initially there are 700 prey and 400 predators.

   (a)

   Find $x$ and $y$ in terms of $t$.

   [6 marks]

   (b)

   (i) Show that the number of prey is always greater than the number of predators.
   (ii) Find the time when the number of prey first reaches 1000.
   (iii) State one limitation of the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions