Particular integralsAQA A-Level Further Maths: Subtopic test

1. 1
   $\frac{d^2y}{dx^2}-\frac{dy}{dx}-6y=12x$.

   (a)

   Which is a suitable trial function for a particular integral?

   [1 mark]

   • A$y=px$
   • B$y=px+q$
   • C$y=px^2+q$
   • D$y=pe^{12x}$

   (b)

   Find the complementary function.

   [1 mark]

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

   (c)

   Find a particular integral.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\frac{d^2y}{dx^2}-4\frac{dy}{dx}+3y=16e^{-x}$.

   (a)

   Which is a suitable trial function for a particular integral?

   [1 mark]

   • A$y=pe^{x}$
   • B$y=pxe^{-x}$
   • C$y=p+qx$
   • D$y=pe^{-x}$

   (b)

   Find the value of $p$ in the particular integral $y=pe^{-x}$.

   [1 mark]

   • A$2$
   • B$-2$
   • C$8$
   • D$\frac{16}{5}$

   (c)

   Find the general solution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=10\cos x$.

   (a)

   Find a particular integral.

   [3 marks]

   (b)

   Given that $y=2$ and $\frac{dy}{dx}=1$ when $x=0$, find $y$ in terms of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\frac{d^2y}{dx^2}-3\frac{dy}{dx}+2y=4e^{x}$.

   (a)

   (i) Explain why $y=pe^{x}$ cannot be a particular integral.
   (ii) Find a particular integral of the form $y=pxe^{x}$.

   [6 marks]

   (b)

   Given that $y=1$ and $\frac{dy}{dx}=0$ when $x=0$, find $y$ in terms of $x$, and find the exact value of $y$ when $x=\ln2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions