Second order equations with constant coefficientsAQA A-Level Further Maths: Subtopic test

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

   (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)

   Find the general solution.

   [1 mark]

   • A$y=Ae^{-2x}+Be^{-3x}$
   • B$y=Ae^{2x}+Bxe^{3x}$
   • C$y=(A+Bx)e^{5x}$
   • 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
   $\frac{d^2y}{dx^2}+4\frac{dy}{dx}+4y=0$.

   (a)

   Find the root(s) of the auxiliary equation.

   [1 mark]

   • A$m=2$ (repeated)
   • B$m=-2$ (repeated)
   • C$m=2$ and $m=-2$
   • D$m=0$ and $m=-4$

   (b)

   Find the general solution.

   [1 mark]

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

   (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
   $\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=0$.

   (a)

   Find the general solution.

   [3 marks]

   (b)

   Given that $y=1$ and $\frac{dy}{dx}=3$ when $x=0$, find $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A function $y(x)$ satisfies $\frac{d^2y}{dx^2}+k\frac{dy}{dx}+9y=0$, where $k$ is a positive constant.

   (a)

   By considering the discriminant of the auxiliary equation, describe the form of the general solution in each of the cases $k>6$, $k=6$ and $0<k<6$, and state what happens to $y$ as $x\to\infty$.

   [6 marks]

   (b)

   Given that $k=10$, $y=4$ and $\frac{dy}{dx}=-2$ when $x=0$, find $y$ in terms of $x$, and state the limit of $y$ as $x\to\infty$.

   [6 marks]

   Total for question 4: 12 marks

End of questions