Coupled first order equationsAQA A-Level Further Maths: Subtopic test

1. 1
   The functions $x$ and $y$ of $t$ satisfy $\frac{dx}{dt}=3x+2y$ and $\frac{dy}{dt}=x+2y$.

   (a)

   Eliminating $y$ gives a second order differential equation for $x$. Which is it?

   [1 mark]

   • A$x''-3x'+2x=0$
   • B$x''-5x'+4x=0$
   • C$x''+5x'+4x=0$
   • D$x''-5x'+6x=0$

   (b)

   Which is the general solution for $x$?

   [1 mark]

   • A$Ae^{-t}+Be^{-4t}$
   • B$Ae^{5t}+Be^{4t}$
   • C$(A+Bt)e^{2t}$
   • D$Ae^{t}+Be^{4t}$

   (c)

   Given that $x=Ae^{t}+Be^{4t}$, find $y$ in terms of $t$, $A$ and $B$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   What does the term $-3y$ in $\frac{dx}{dt}$ represent?

   [1 mark]

   • APrey grow faster when there are more predators
   • BPredators grow faster when there are more prey
   • CPredators reduce the prey at a rate proportional to the number of predators
   • DPredators die at a rate proportional to their number

   (b)

   When $x=0$ and $y=1$, find $\frac{dx}{dt}$ and $\frac{dy}{dt}$.

   [1 mark]

   • A$-3$ and $-2$
   • B$-3$ and $2$
   • C$3$ and $-2$
   • D$0$ and $0$

   (c)

   Show that $x''-x=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 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 $\frac{dy}{dt}=2x-3y$.

   (a)

   Show that $x''+5x'+4x=0$.

   [3 marks]

   (b)

   Initially tank $P$ holds 3 kg of salt and tank $Q$ holds none. Find $x$ and $y$ in terms of $t$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a model of a predator-prey system, $x$ and $y$ measure the departures, in thousands, of the prey and predator populations from their equilibrium levels at time $t$ years, where $\frac{dx}{dt}=-x-2y$ and $\frac{dy}{dt}=2x-y$.

   (a)

   (i) Show that $x''+2x'+5x=0$.
   (ii) Given that $x=4$ and $y=0$ when $t=0$, find $x$ in terms of $t$.

   [6 marks]

   (b)

   Given that $x=4e^{-t}\cos2t$ and $y=4e^{-t}\sin2t$:
   (i) find the first time $t>0$ at which the prey population is at its equilibrium level;
   (ii) find the predator departure at this time;
   (iii) describe the long-term behaviour of the two populations.

   [6 marks]

   Total for question 4: 12 marks

End of questions