Separable first order differential equationsEdexcel International A Level Maths: Subtopic test

1. 1
   A curve satisfies the differential equation $\frac{dy}{dx}=\frac{x}{y}$ with $y>0$.

   (a)

   Which of the following is the general solution of the differential equation?

   [1 mark]

   • A$\ln y=\ln x+A$
   • B$y=\frac{x^2}{2}+A$
   • C$y^2=\frac{x^2}{2}+A$
   • D$y^2=x^2+A$

   (b)

   Given also that $y=3$ when $x=2$, find the value of $A$ in the solution $y^2=x^2+A$.

   [1 mark]

   • A$5$
   • B$-5$
   • C$13$
   • D$1$

   (c)

   Given also that $y=3$ when $x=2$, find the value of $y$ when $x=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Water drains from a tank. The depth $h$ metres of water at time $t$ minutes satisfies $\frac{dh}{dt}=-k\sqrt{h}$, where $k$ is a positive constant. Initially $h=4$, and after 10 minutes $h=1$.

   (a)

   Which of the following is the general solution of the differential equation?

   [1 mark]

   • A$\sqrt{h}=-kt+c$
   • B$2\sqrt{h}=-kt+c$
   • C$\frac{2}{3}h^{3/2}=-kt+c$
   • D$\ln h=-kt+c$

   (b)

   Find the value of $k$.

   [1 mark]

   • A$0.1$
   • B$0.3$
   • C$0.2$
   • D$0.4$

   (c)

   Find the time taken for the tank to empty completely.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ satisfies $\frac{dy}{dx}=\frac{2y+1}{x}$ for $x>0$, and passes through the point $(1,2)$.

   (a)

   Find the general solution of the differential equation, giving your answer in the form $\ln|2y+1|=f(x)$.

   [3 marks]

   (b)

   Hence find the equation of $C$, giving $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cup of coffee cools in a room at a constant temperature of $20^\circ$C. Its temperature $\theta^\circ$C at time $t$ minutes satisfies $\frac{d\theta}{dt}=-k(\theta-20)$, where $k$ is a positive constant. Initially $\theta=90$, and after 5 minutes $\theta=70$. A calculator may be used.

   (a)

   Solve the differential equation to show that $\theta=20+70e^{-kt}$, and find the exact value of $k$.

   [6 marks]

   (b)

   The coffee can be drunk once its temperature has fallen to $60^\circ$C.
   (i) Find the time at which this happens.
   (ii) Find the rate at which the temperature is falling at that moment.
   (iii) State the temperature the model predicts in the long term, and comment on whether this is realistic.

   [6 marks]

   Total for question 4: 12 marks

End of questions