Second-order linear differential equationsEdexcel A-Level Further Maths: Subtopic test

1. 1
   Consider the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}-8y=0$.

   (a)

   Find the roots of the auxiliary equation.

   [1 mark]

   • A$m=2$ and $m=-4$
   • B$m=-2$ and $m=4$
   • C$m=2$ and $m=4$
   • D$m=8$ and $m=-1$

   (b)

   Find the general solution of the differential equation.

   [1 mark]

   • A$y=Ae^{-2x}+Be^{4x}$
   • B$y=Ae^{2x}+Be^{-4x}$
   • C$y=(A+Bx)e^{2x}$
   • D$y=A\cos2x+B\sin4x$

   (c)

   Given that $y=0$ and $\frac{dy}{dx}=6$ when $x=0$, find the particular solution.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the differential equation $\frac{d^2y}{dx^2}+4\frac{dy}{dx}+13y=0$.

   (a)

   Find the roots of the auxiliary equation.

   [1 mark]

   • A$m=-4\pm6\mathrm{i}$
   • B$m=2\pm3\mathrm{i}$
   • C$m=-2\pm3\mathrm{i}$
   • D$m=1$ and $m=-5$

   (b)

   Find the general solution of the differential equation.

   [1 mark]

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

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $\frac{d^2y}{dx^2}-3\frac{dy}{dx}+2y=4x+2$.

   (a)

   Find a particular integral.

   [3 marks]

   (b)

   Given that $y=7$ and $\frac{dy}{dx}=7$ when $x=0$, use your answer to part (a) to find $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the differential equation $\frac{d^2y}{dx^2}-5\frac{dy}{dx}+6y=4e^{3x}$.

   (a)

   (i) Find the complementary function.
   (ii) Explain why a particular integral of the form $y=ke^{3x}$ cannot be used.
   (iii) Find a particular integral.

   [6 marks]

   (b)

   Given that $y=1$ and $\frac{dy}{dx}=0$ when $x=0$, find $y$ in terms of $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions