Core Pure: Differential equationsEdexcel A-Level Further Maths: Topic test

1. 1
   Consider the differential equation $\frac{dy}{dx}+\frac{2x}{1+x^2}y=3x$.

   (a)

   Find the integrating factor.

   [1 mark]

   • A$\ln(1+x^2)$
   • B$1+x^2$
   • C$\frac{2x}{1+x^2}$
   • D$\mathrm{e}^{2x}$

   (b)

   Which expression is the general solution?

   [1 mark]

   • A$y=\dfrac{\frac32x^2+\frac34x^4+c}{1+x^2}$
   • B$y=\frac32x^2+\frac34x^4+c$
   • C$y=\dfrac{\frac32x^2+c}{1+x^2}$
   • D$y=\left(\frac32x^2+\frac34x^4+c\right)(1+x^2)$

   (c)

   Given that $y=1$ when $x=0$, find $y$ in terms of $x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A buoy floats on a calm sea and bobs vertically. Its displacement $x$ metres above its equilibrium level at time $t$ seconds satisfies $\frac{d^2x}{dt^2}=-16x$.

   (a)

   What is the period of the motion?

   [1 mark]

   • A$\frac{\pi}{8}$ s
   • B$4$ s
   • C$8\pi$ s
   • D$\frac{\pi}{2}$ s

   (b)

   Which expression is the general solution of the equation?

   [1 mark]

   • A$x=A\cos16t+B\sin16t$
   • B$x=A\mathrm{e}^{4t}+B\mathrm{e}^{-4t}$
   • C$x=A\cos4t+B\sin4t$
   • D$x=(A+Bt)\mathrm{e}^{4t}$

   (c)

   Given that $x=0.3$ and $\frac{dx}{dt}=2.4$ when $t=0$, find the amplitude of the motion.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $\frac{d^2y}{dx^2}+6\frac{dy}{dx}+9y=5\mathrm{e}^{-x}$.

   (a)

   Find the complementary function.

   [3 marks]

   (b)

   Find the particular solution for which $y=1$ and $\frac{dy}{dx}=0$ when $x=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A mass on a spring is driven by an external force, so that its displacement $x$ metres from equilibrium at time $t$ seconds satisfies $\frac{d^2x}{dt^2}+2\frac{dx}{dt}+5x=17\cos2t$.

   (a)

   Find the general solution of the differential equation.

   [6 marks]

   (b)

   (i) Given that $x=1$ and $\frac{dx}{dt}=0$ when $t=0$, find $x$ in terms of $t$. [4]
   (ii) Describe the long-term motion of the mass and state its amplitude. [2]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The concentration $C$ mg per litre of a drug in a patient's blood at time $t$ hours after an injection satisfies $\frac{dC}{dt}+0.4C=6\mathrm{e}^{-0.1t}$.

   (a)

   Find the integrating factor.

   [1 mark]

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

   (b)

   After multiplying the equation by the integrating factor, what is the right-hand side?

   [1 mark]

   • A$6\mathrm{e}^{-0.5t}$
   • B$6\mathrm{e}^{0.5t}$
   • C$6\mathrm{e}^{-0.1t}$
   • D$6\mathrm{e}^{0.3t}$

   (c)

   Given that $C=0$ when $t=0$, show that $C=20\left(\mathrm{e}^{-0.1t}-\mathrm{e}^{-0.4t}\right)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A vehicle's suspension is modelled so that the vertical displacement $x$ metres of the body from equilibrium at time $t$ seconds satisfies $\frac{d^2x}{dt^2}+10\frac{dx}{dt}+16x=0$.

   (a)

   Which statement describes the damping?

   [1 mark]

   • AHeavy damping, because the auxiliary equation has two distinct real roots
   • BCritical damping, because the auxiliary equation has a repeated root
   • CLight damping, because the auxiliary equation has complex roots
   • DNo damping, because the equation is homogeneous

   (b)

   Which expression is the general solution?

   [1 mark]

   • A$x=A\mathrm{e}^{2t}+B\mathrm{e}^{8t}$
   • B$x=A\mathrm{e}^{-2t}+B\mathrm{e}^{-8t}$
   • C$x=(A+Bt)\mathrm{e}^{-2t}$
   • D$x=\mathrm{e}^{-5t}(A\cos3t+B\sin3t)$

   (c)

   Given that $x=0.5$ and $\frac{dx}{dt}=0$ when $t=0$, find $x$ in terms of $t$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The temperature deviations $x$ and $y$ (in $^\circ$C) of two connected rooms from their target temperature at time $t$ hours satisfy $\frac{dx}{dt}=x-2y$ and $\frac{dy}{dt}=3x-4y$.

   (a)

   Show that $\frac{d^2x}{dt^2}+3\frac{dx}{dt}+2x=0$.

   [3 marks]

   (b)

   Given that $x=3$ and $y=2$ when $t=0$, find $x$ and $y$ in terms of $t$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A model for a damped system has $x$ and $y$ satisfying $\frac{dx}{dt}=y$ and $\frac{dy}{dt}=-9x-6y$ for $t\ge0$, with $x=2$ and $y=4$ when $t=0$.

   (a)

   Show that $\frac{d^2x}{dt^2}+6\frac{dx}{dt}+9x=0$, and hence find $x$ and $y$ in terms of $t$.

   [6 marks]

   (b)

   The model describes the displacement $x$ of a damped oscillator.
   (i) Explain why the system is critically damped. [2]
   (ii) Find the maximum value of $x$ for $t\ge0$, and the time at which it occurs. [4]

   [6 marks]

   Total for question 8: 12 marks

End of questions