First order linear differential equationsEdexcel International A Level Further Maths: Subtopic test

1. 1
   Consider the differential equation $\frac{dy}{dx}+3y=e^{2x}$.

   (a)

   Find the integrating factor.

   [1 mark]

   • A$e^{2x}$
   • B$3x$
   • C$e^{3x}$
   • D$e^{-3x}$

   (b)

   After multiplying the equation by the integrating factor, which statement is correct?

   [1 mark]

   • A$\frac{d}{dx}\left(ye^{3x}\right)=e^{2x}$
   • B$\frac{d}{dx}\left(ye^{3x}\right)=e^{5x}$
   • C$\frac{d}{dx}\left(ye^{3x}\right)=e^{-x}$
   • D$\frac{d}{dx}\left(ye^{2x}\right)=e^{5x}$

   (c)

   Find the general solution.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the differential equation $x\frac{dy}{dx}-2y=x^3$ for $x>0$.

   (a)

   Which equation is the given equation written in the form $\frac{dy}{dx}+Py=Q$?

   [1 mark]

   • A$\frac{dy}{dx}-2y=x^3$
   • B$\frac{dy}{dx}-\frac2xy=x^3$
   • C$\frac{dy}{dx}+\frac2xy=x^2$
   • D$\frac{dy}{dx}-\frac2xy=x^2$

   (b)

   Find the integrating factor.

   [1 mark]

   • A$\frac{1}{x^2}$
   • B$x^2$
   • C$e^{-2x}$
   • D$-2\ln x$

   (c)

   Find the general solution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $\frac{dy}{dx}+y\tan x=\sec x$ for $0<x<\frac{\pi}{2}$.

   (a)

   Show that the integrating factor is $\sec x$.

   [3 marks]

   (b)

   Given that $y=1$ when $x=0$, find $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A tank initially contains $100$ litres of pure water. Brine containing $0.5$ kg of salt per litre flows into the tank at $4$ litres per minute. The mixture is kept uniform by stirring and flows out at $4$ litres per minute, so the volume stays at $100$ litres. At time $t$ minutes the tank contains $m$ kg of salt.

   (a)

   Show that $\frac{dm}{dt}+\frac{m}{25}=2$ and hence find $m$ in terms of $t$.

   [6 marks]

   (b)

   (i) Find, to the nearest $0.1$ minute, the time when the concentration of salt in the tank first reaches $0.25$ kg per litre.
   (ii) Describe what happens to the mass and the concentration of salt in the long term.
   (iii) Suggest one limitation of the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions