Differential equationsAQA A-Level Further Maths: Topic test

1. 1
   A differential equation is $x\dfrac{dy}{dx}+2y=6x^2$, for $x>0$.

   (a)

   What is an integrating factor for this equation?

   [1 mark]

   • A$x$
   • B$x^2$
   • C$e^{2x}$
   • D$2\ln x$

   (b)

   After multiplying by the integrating factor, which equation results?

   [1 mark]

   • A$\frac{d}{dx}\left(x^2y\right)=6x^2$
   • B$\frac{d}{dx}\left(x^2y\right)=12x$
   • C$\frac{d}{dx}(xy)=6x^3$
   • D$\frac{d}{dx}\left(x^2y\right)=6x^3$

   (c)

   Find the general solution for $y$ in terms of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function $y(x)$ satisfies $\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+13y=0$.

   (a)

   What is the auxiliary equation?

   [1 mark]

   • A$m^2-6m+13=0$
   • B$m^2+6m+13=0$
   • C$m^2-6m=13$
   • D$m^2-6m-13=0$

   (b)

   Which is the general solution?

   [1 mark]

   • A$y=e^{-3x}\left(A\cos2x+B\sin2x\right)$
   • B$y=e^{3x}\left(A\cos13x+B\sin13x\right)$
   • C$y=e^{3x}\left(A\cos2x+B\sin2x\right)$
   • D$y=e^{2x}\left(A\cos3x+B\sin3x\right)$

   (c)

   Given that $y=1$ and $\frac{dy}{dx}=5$ when $x=0$, find the particular solution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A differential equation is $\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}+3y=3e^{-x}$.

   (a)

   Find the complementary function, and explain why $y=ae^{-x}$ cannot be used as a particular integral.

   [3 marks]

   (b)

   Find the general solution of the differential equation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.5$ kg lies on a smooth horizontal table and is attached to one end of a light spring whose other end is fixed. When $P$ has displacement $x$ metres from its equilibrium position, the spring exerts a force of magnitude $32|x|$ newtons directed towards the equilibrium position. At time $t=0$, $P$ is released from rest with $x=0.3$.

   (a)

   Show that $P$ moves with simple harmonic motion, find $x$ in terms of $t$, and find the period of the motion.

   [6 marks]

   (b)

   The particle now also experiences a resistive force of magnitude $4|x'|$ newtons. Show that $x''+8x'+64x=0$, find the general solution, and state, with a reason, the type of damping.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A drug is infused into a patient's bloodstream at a constant rate. The amount $x$ mg of the drug in the bloodstream $t$ hours after the infusion begins satisfies $\dfrac{dx}{dt}+0.4x=6$, with $x=0$ when $t=0$.

   (a)

   What is an integrating factor for this equation?

   [1 mark]

   • A$e^{0.4}$
   • B$e^{-0.4t}$
   • C$e^{0.4t}$
   • D$0.4t$

   (b)

   What value does $x$ approach as $t\to\infty$?

   [1 mark]

   • A$15$
   • B$6$
   • C$2.4$
   • D$0.4$

   (c)

   Find $x$ in terms of $t$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A function $y(x)$ satisfies $\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}-5y=0$.

   (a)

   What are the roots of the auxiliary equation?

   [1 mark]

   • A$m=-5$ and $m=1$
   • B$m=4$ and $m=-5$
   • C$m=-1$ and $m=-5$
   • D$m=5$ and $m=-1$

   (b)

   Given that $y=2$ and $\frac{dy}{dx}=-2$ when $x=0$, which is the particular solution?

   [1 mark]

   • A$y=e^{5x}+e^{-x}$
   • B$y=2e^{-x}$
   • C$y=2e^{5x}$
   • D$y=3e^{-x}-e^{5x}$

   (c)

   Find the range of values of $k$ for which the solutions of $\frac{d^2y}{dx^2}-4\frac{dy}{dx}+ky=0$ are oscillatory.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A differential equation is $\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}+5y=8\cos x$.

   (a)

   Find a particular integral of the form $y=a\cos x+b\sin x$.

   [3 marks]

   (b)

   Given that $y=0$ and $\frac{dy}{dx}=3$ when $x=0$, find $y$ in terms of $x$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The areas $x$ cm$^2$ and $y$ cm$^2$ of two mutually supporting colonies of fungus on a leaf, $t$ days after they are first measured, are modelled by $\dfrac{dx}{dt}=x+2y$ and $\dfrac{dy}{dt}=2x+y$. When $t=0$, $x=5$ and $y=1$.

   (a)

   Show that $\frac{d^2x}{dt^2}-2\frac{dx}{dt}-3x=0$, and hence find the general solution for $x$.

   [6 marks]

   (b)

   Find $x$ and $y$ in terms of $t$, and describe the long-term ratio of $x$ to $y$.

   [6 marks]

   Total for question 8: 12 marks

End of questions