Separable differential equations and families of solutionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A curve $C$ passes through the point $(0,3)$ and satisfies the differential equation $\frac{dy}{dx}=2xy$, where $y>0$.

   (a)

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

   [1 mark]

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

   (b)

   Find the value of $y$ on $C$ when $x=1$.

   [1 mark]

   • A$e^3$
   • B$3+e$
   • C$3e^2$
   • D$3e$

   (c)

   Show that $C$ has a minimum point at $(0,3)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A drink cools in a room at a constant temperature of $20\,^\circ\text{C}$. At time $t$ minutes its temperature is $\theta\,^\circ\text{C}$, and the rate of decrease of $\theta$ is proportional to $\theta-20$, with constant of proportionality $k>0$. Initially $\theta=80$.

   (a)

   Which differential equation models the cooling?

   [1 mark]

   • A$\frac{d\theta}{dt}=k(\theta-20)$
   • B$\frac{d\theta}{dt}=-k\theta$
   • C$\frac{d\theta}{dt}=-k(\theta-20)$
   • D$\frac{d\theta}{dt}=-\frac{k}{\theta-20}$

   (b)

   Solve the differential equation to find $\theta$ in terms of $t$ and $k$.

   [1 mark]

   • A$\theta=20+80e^{-kt}$
   • B$\theta=20+60e^{-kt}$
   • C$\theta=60+20e^{-kt}$
   • D$\theta=20+60e^{kt}$

   (c)

   After $5$ minutes the temperature is $50\,^\circ\text{C}$. Find the exact value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve satisfies the differential equation $\frac{dy}{dx}=xe^{-y}$.

   (a)

   Find the general solution, giving $e^y$ in terms of $x$.

   [3 marks]

   (b)

   Given that $y=0$ when $x=0$, find $y$ in terms of $x$. State the equation of the line of symmetry of the curve and the coordinates of its minimum point.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ passes through the point $(1,1)$ and, for $x>0$, satisfies the differential equation $x\frac{dy}{dx}=y^2$.

   (a)

   Solve the differential equation to find $y$ in terms of $x$.

   [6 marks]

   (b)

   Describe the graph of $y=\frac{1}{1-\ln x}$ for $x>0$, $x\neq e$. Give the gradient at $(1,1)$, the equations of both asymptotes, where $y$ is positive and where it is negative, and the behaviour as $x\to0^+$ and as $x\to e$ from either side.

   [6 marks]

   Total for question 4: 12 marks

End of questions