Series solutions and reducible differential equationsEdexcel A-Level Further Maths: Mind map

Exam questions on Series solutions and reducible differential equations

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 $x=0$.
   Use your answer to part (b) to find the value of $\dfrac{\mathrm{d}^4y}{\mathrm{d}x^4}$ at $x=0$.2 marks
2. The function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=x-y^2$, with $y=1$ at $x=0$.
   Hence find the first four terms of the series solution of the differential equation in ascending powers of $x$.2 marks
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 $z=\dfrac1y$.
   Show that the substitution transforms the equation into $\dfrac{\mathrm{d}z}{\mathrm{d}x}-\dfrac{z}{x}=-1$.3 marks