Separable differential equationsEdexcel A-Level Maths: Subtopic test

1. 1
   A curve satisfies the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=2xy$ with $y>0$.

   (a)

   Which line correctly separates the variables?

   [1 mark]

   • A$\int y\,\mathrm{d}y=\int2x\,\mathrm{d}x$
   • B$\int\frac1x\,\mathrm{d}y=\int2y\,\mathrm{d}x$
   • C$\int\frac1y\,\mathrm{d}y=\int2x\,\mathrm{d}x$
   • D$\int\frac1y\,\mathrm{d}y=\int2xy\,\mathrm{d}x$

   (b)

   Find the general solution.

   [1 mark]

   • A$y=Ae^{x^2}$
   • B$y=e^{x^2}+c$
   • C$y=Ax^2$
   • D$y=x^2+\ln A$

   (c)

   Given that $y=3$ when $x=0$, find the exact value of $y$ when $x=1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass $m$ grams of a radioactive sample at time $t$ years satisfies $\frac{\mathrm{d}m}{\mathrm{d}t}=-0.05m$. Initially the mass is $80$ g.

   (a)

   Find the general solution for $m$ in terms of $t$.

   [1 mark]

   • A$m=Ae^{0.05t}$
   • B$m=-0.05t+c$
   • C$m=e^{-0.05t}+c$
   • D$m=Ae^{-0.05t}$

   (b)

   Find the mass remaining after $10$ years.

   [1 mark]

   • A$40.0$ g
   • B$48.5$ g
   • C$0.539$ g
   • D$132$ g

   (c)

   Find the time taken for the mass to fall to $40$ g, giving your answer in years to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle moves in a straight line. Its velocity $v$ m s$^{-1}$ at time $t$ seconds satisfies $\frac{\mathrm{d}v}{\mathrm{d}t}=-0.4v^2$, and $v=5$ when $t=0$.

   (a)

   Show that $v=\frac{5}{2t+1}$.

   [3 marks]

   (b)

   Find the exact distance travelled in the first $4$ seconds. Explain why the model is unrealistic for large values of $t$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Water drains from a tank. The depth of water, $h$ metres, at time $t$ minutes satisfies $\frac{\mathrm{d}h}{\mathrm{d}t}=-0.1\sqrt{h}$ for $h>0$. Initially the depth is $4$ m.

   (a)

   Solve the differential equation to find $h$ in terms of $t$, and state the time at which the tank is empty. State the range of values of $t$ for which your solution is valid.

   [6 marks]

   (b)

   A second tank is modelled by the same differential equation but starts with depth $h_0$ metres. A student claims that doubling the initial depth doubles the time taken to empty. Use the model to evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions