FP2: First order differential equationsEdexcel International A Level Further Maths: Topic test

1. 1
   Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{3y}{x}=4x$ for $x>0$.

   (a)

   Find an integrating factor for the differential equation.

   [1 mark]

   • A$\mathrm{e}^{3x}$
   • B$x^3$
   • C$3\ln x$
   • D$x^{-3}$

   (b)

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

   [1 mark]

   • A$y=\frac45x^2+C$
   • B$y=\frac45x^2+Cx^{3}$
   • C$y=4x^2+Cx^{-3}$
   • D$y=\frac45x^2+Cx^{-3}$

   (c)

   Find the particular solution for which $y=1$ when $x=1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass $m$ kg of a radioactive sample at time $t$ years decreases at a rate proportional to $m$. Initially $m=80$, and after $10$ years $m=40$.

   (a)

   Which differential equation models the mass of the sample, where $k$ is a positive constant?

   [1 mark]

   • A$\frac{\mathrm{d}m}{\mathrm{d}t}=km$
   • B$\frac{\mathrm{d}m}{\mathrm{d}t}=-k$
   • C$\frac{\mathrm{d}m}{\mathrm{d}t}=-km$
   • D$\frac{\mathrm{d}m}{\mathrm{d}t}=-\frac km$

   (b)

   Find the mass of the sample after $30$ years.

   [1 mark]

   • A$10$ kg
   • B$20$ kg
   • C$0$ kg
   • D$26.7$ kg

   (c)

   Find the value of $k$ in the differential equation, giving your answer in exact form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $x^2\frac{\mathrm{d}y}{\mathrm{d}x}=y^2+2xy$ for $x>0$, and the substitution $y=vx$, where $v$ is a function of $x$.

   (a)

   Show that the substitution transforms the differential equation into $x\frac{\mathrm{d}v}{\mathrm{d}x}=v(v+1)$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drug is administered so that the concentration $c$ mg per litre in a patient's blood at time $t$ hours satisfies $\frac{\mathrm{d}c}{\mathrm{d}t}+0.4c=12\mathrm{e}^{-0.1t}$, with $c=0$ when $t=0$.

   (a)

   Solve the differential equation to find $c$ in terms of $t$.

   [6 marks]

   (b)

   Find the time at which the concentration is greatest, justifying that it is a maximum, and find the maximum concentration. Give your answers to $3$ significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=\cos x\,(1+y^2)$, with $y=0$ when $x=0$.

   (a)

   Which of the following is obtained by separating the variables?

   [1 mark]

   • A$\int\frac{1}{1+y^2}\,\mathrm{d}y=\int\cos x\,\mathrm{d}x$
   • B$\int(1+y^2)\,\mathrm{d}y=\int\cos x\,\mathrm{d}x$
   • C$\int\frac{1}{y^2}\,\mathrm{d}y=\int\cos x\,\mathrm{d}x$
   • D$\int\frac{1}{1+y^2}\,\mathrm{d}y=\int\sec x\,\mathrm{d}x$

   (b)

   Which of the following is the particular solution?

   [1 mark]

   • A$y=\sin x$
   • B$y=\sin(\tan x)$
   • C$y=\tan(\sin x)$
   • D$y=\tan^{-1}(\sin x)$

   (c)

   Find the value of $y$ when $x=\frac\pi2$, giving your answer to $3$ significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=\cos(x+y)$ and the substitution $z=x+y$.

   (a)

   Which of the following is $\frac{\mathrm{d}z}{\mathrm{d}x}$ in terms of $z$?

   [1 mark]

   • A$\cos z$
   • B$1-\cos z$
   • C$-\sin z$
   • D$1+\cos z$

   (b)

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

   [1 mark]

   • A$\tan(x+y)=x+C$
   • B$\tan\frac{x+y}{2}=x+C$
   • C$\sin\frac{x+y}{2}=x+C$
   • D$\tan\frac{x+y}{2}=\frac x2+C$

   (c)

   Find the particular solution for which $y=0$ when $x=0$, giving $y$ in terms of $x$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{4y}{2x+1}$ for $x>-\frac12$ and $y>0$.

   (a)

   Find the general solution of the differential equation.

   [3 marks]

   (b)

   A solution curve passes through the point $(0,3)$. Find its equation, and find the area of the region bounded by the curve, the $x$-axis, the $y$-axis and the line $x=1$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Consider the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x}+y=xy^3$, where $y>0$, and the substitution $z=\frac{1}{y^2}$.

   (a)

   Show that the substitution transforms the differential equation into $\frac{\mathrm{d}z}{\mathrm{d}x}-2z=-2x$, and hence find the general solution for $z$ in terms of $x$.

   [6 marks]

   (b)

   Given that $y=1$ when $x=0$, find $y$ in terms of $x$, and describe the behaviour of $y$ as $x\to\infty$.

   [6 marks]

   Total for question 8: 12 marks

End of questions