Constant coefficient second order equationsEdexcel International A Level Further Maths: Subtopic test

1. 1
   Consider the differential equation $\frac{d^2y}{dx^2}-5\frac{dy}{dx}+6y=0$.

   (a)

   Find the roots of the auxiliary equation.

   [1 mark]

   • A$m=-2$ and $m=-3$
   • B$m=2$ and $m=3$
   • C$m=1$ and $m=6$
   • D$m=-1$ and $m=-6$

   (b)

   Which of the following is the general solution?

   [1 mark]

   • A$y=Ae^{-2x}+Be^{-3x}$
   • B$y=(A+Bx)e^{2x}$
   • C$y=Ae^{5x}+Be^{6x}$
   • D$y=Ae^{2x}+Be^{3x}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the differential equation $\frac{d^2y}{dx^2}+6\frac{dy}{dx}+9y=0$.

   (a)

   Which statement about the roots of the auxiliary equation is correct?

   [1 mark]

   • AThe roots are $m=-1$ and $m=-9$
   • BThere is a repeated root $m=3$
   • CThere is a repeated root $m=-3$
   • DThe roots are complex, $m=-3\pm3i$

   (b)

   Which of the following is the general solution?

   [1 mark]

   • A$y=(A+Bx)e^{-3x}$
   • B$y=Ae^{-3x}+Be^{3x}$
   • C$y=(A+Bx)e^{3x}$
   • D$y=e^{-3x}(A\cos3x+B\sin3x)$

   (c)

   Find the particular solution for which $y=2$ and $\frac{dy}{dx}=0$ when $x=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=5x+7$.

   (a)

   Find the complementary function.

   [3 marks]

   (b)

   Find a particular integral of the form $y=px+q$, and hence write down the general solution.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the differential equation $\frac{d^2y}{dx^2}+4y=\sin2x$.

   (a)

   Find the general solution.

   [6 marks]

   (b)

   Given that $y=0$ and $\frac{dy}{dx}=1$ when $x=0$, find the particular solution. Describe the behaviour of this solution for large positive $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions