Separable differential equationsAQA A-Level Maths: Subtopic test

1. 1
   A curve satisfies the differential equation $\frac{dy}{dx}=2xy$, where $y>0$.

   (a)

   Which equation is obtained by separating the variables?

   [1 mark]

   • A$\int y\,dy=\int2x\,dx$
   • B$\int dy=\int2xy\,dx$
   • C$\int\frac1y\,dy=\int2x\,dx$
   • D$\int\frac1y\,dy=\int2\,dx$

   (b)

   Which expression is the general solution of the differential equation?

   [1 mark]

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

   (c)

   Given that $y=3$ when $x=0$, find $y$ in terms of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves in a straight line. Its velocity $v$ m s$^{-1}$ at time $t$ seconds satisfies $\frac{dv}{dt}=-0.5v$, and $v=8$ when $t=0$.

   (a)

   Which expression gives $v$ in terms of $t$?

   [1 mark]

   • A$v=8e^{-0.5t}$
   • B$v=8e^{0.5t}$
   • C$v=8-0.5t$
   • D$v=7+e^{-0.5t}$

   (b)

   Find the velocity when $t=4$.

   [1 mark]

   • A$0.147$ m s$^{-1}$
   • B$1.08$ m s$^{-1}$
   • C$4.85$ m s$^{-1}$
   • D$6$ m s$^{-1}$

   (c)

   Explain, using your solution, why the model predicts that the particle never comes to rest, and suggest why this is unrealistic.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve satisfies the differential equation $\frac{dy}{dx}=6x+3xy$ for $x\ge0$.

   (a)

   Show that the general solution can be written $y=Ae^{\frac32x^2}-2$, where $A$ is a constant.

   [3 marks]

   (b)

   The curve passes through the point $(0,1)$. Find the value of $x$ for which $y=10$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cup of tea 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 and $\theta>20$. Initially $\theta=90$, and after 5 minutes $\theta=60$.

   (a)

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

   [6 marks]

   (b)

   The tea is cool enough to drink when $\theta=50$. Find the time at which this happens, to the nearest 0.1 minute. State the temperature that the model predicts in the long term, and give one reason why the model may be inaccurate.

   [6 marks]

   Total for question 4: 12 marks

End of questions