First-order differential equations and integrating factorsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Consider the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^2$ for $x>0$.

   (a)

   Find an integrating factor for this equation.

   [1 mark]

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

   (b)

   Find the general solution of the differential equation.

   [1 mark]

   • A$y=\frac{x^3}{5}+c$
   • B$y=\frac{x^3}{5}+\frac{c}{x^2}$
   • C$y=\frac{x^5}{5}+\frac{c}{x^2}$
   • D$y=x^2\left(\frac{x^5}{5}+c\right)$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve satisfies $\frac{dy}{dx}+y\tan x=\cos x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

   (a)

   Find an integrating factor for this equation.

   [1 mark]

   • A$\sec x$
   • B$\cos x$
   • C$\ln\sec x$
   • D$\sec^2x$

   (b)

   Find the general solution of the differential equation.

   [1 mark]

   • A$y=(x+c)\sec x$
   • B$y=x\cos x+c$
   • C$y=(x+c)\cos x$
   • D$y=x\sec x+c$

   (c)

   The curve passes through the point $\left(\frac{\pi}{3},\frac{\pi}{3}\right)$. Find $y$ in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A tank holds 200 litres of well-stirred brine containing 10 kg of salt at time $t=0$, where $t$ is in minutes. Brine of concentration 0.5 kg per litre flows in at 4 litres per minute, and the well-stirred mixture flows out at 4 litres per minute. Let $m$ kg be the mass of salt in the tank at time $t$.

   (a)

   Show that $\frac{dm}{dt}+\frac{m}{50}=2$.

   [3 marks]

   (b)

   Solve the differential equation in part (a) to find $m$ in terms of $t$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A parachutist falls vertically from rest. At time $t$ seconds her speed is $v$ m s$^{-1}$ and her motion is modelled by $\frac{dv}{dt}=10-0.4v$.

   (a)

   Use an integrating factor to show that $v=25\left(1-e^{-0.4t}\right)$, and state the speed that the model predicts she approaches.

   [6 marks]

   (b)

   Find
   (i) the distance she falls in the first 5 seconds,
   (ii) the time at which her speed first reaches 90% of the speed in part (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions