Series solutions and reducible differential equationsEdexcel A-Level Further Maths: Subtopic test

1. 1
   The function $y$ satisfies $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\dfrac{\mathrm{d}y}{\mathrm{d}x}+y=0$, with $y=1$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$ at $x=0$.

   (a)

   Find the value of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$ at $x=0$.

   [1 mark]

   • A$1$
   • B$0$
   • C$-1$
   • D$-2$

   (b)

   Which expression gives $\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}$?

   [1 mark]

   • A$-xy''-y'$
   • B$-2y'-xy''$
   • C$-2y'-xy''-y$
   • D$-2xy''-y'$

   (c)

   Use your answer to part (b) to find the value of $\dfrac{\mathrm{d}^4y}{\mathrm{d}x^4}$ at $x=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=x-y^2$, with $y=1$ at $x=0$.

   (a)

   Find the value of $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}$ at $x=0$.

   [1 mark]

   • A$-1$
   • B$1$
   • C$3$
   • D$2$

   (b)

   Find the value of $\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}$ at $x=0$.

   [1 mark]

   • A$-2$
   • B$-6$
   • C$8$
   • D$-8$

   (c)

   Hence find the first four terms of the series solution of the differential equation in ascending powers of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The differential equation $x\dfrac{\mathrm{d}y}{\mathrm{d}x}+y=xy^2$ is to be solved for $x>0$ using the substitution $z=\dfrac1y$.

   (a)

   Show that the substitution transforms the equation into $\dfrac{\mathrm{d}z}{\mathrm{d}x}-\dfrac{z}{x}=-1$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The differential equation $x^2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}-2x\dfrac{\mathrm{d}y}{\mathrm{d}x}+2y=4x^3$ is to be solved for $x>0$ using the substitution $x=\mathrm{e}^t$.

   (a)

   Show that the substitution transforms the equation into $\dfrac{\mathrm{d}^2y}{\mathrm{d}t^2}-3\dfrac{\mathrm{d}y}{\mathrm{d}t}+2y=4\mathrm{e}^{3t}$.

   [6 marks]

   (b)

   Hence find the general solution, giving $y$ in terms of $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions