Series solutions of differential equationsEdexcel International A Level Further Maths: Mind map

Exam questions on Series solutions of differential equations

1. The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=x+2y$, with $y=1$ and $\frac{dy}{dx}=2$ at $x=0$. A series solution in ascending powers of $x$ is to be found.
   Find the series solution up to and including the term in $x^4$, and use it to estimate $y(0.2)$ to 4 decimal places.2 marks
2. The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=xy$, with $y=1$ and $\frac{dy}{dx}=1$ at $x=0$.
   Find the series solution up to and including the term in $x^4$, and use it to estimate $y(0.4)$ to 3 decimal places.2 marks
3. The function $y(x)$ satisfies the first-order differential equation $\frac{dy}{dx}=x^2+y$, with $y=1$ at $x=0$.
   Find the values of $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$ and $\frac{d^3y}{dx^3}$ at $x=0$.3 marks