Numerical solution of differential equationsEdexcel A-Level Further Maths: Mind map

Exam questions on Numerical solution of differential equations

1. The differential equation $\frac{dy}{dx}=x+y$, with $y=1$ when $x=0$, is solved numerically using the approximation $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_n}{h}$ with $h=0.2$. Here $x_n=0.2n$ and $y_n$ is the approximation to $y$ at $x_n$.
   Given that $y_3=1.856$, and that the exact solution is $y=2e^x-x-1$, calculate the percentage error in $y_3$ as an approximation to $y(0.6)$.2 marks
2. The differential equation $\frac{dy}{dx}=x-y$, with $y=1$ when $x=0$, is solved numerically using the approximation $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_{n-1}}{2h}$ with $h=0.1$. Here $x_n=0.1n$, $y_0=1$, and the value $y_1=0.9$ is found from the approximation $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_n}{h}$.
   Find $y_3$.2 marks
3. The function $y$ satisfies $\frac{d^2y}{dx^2}=-y$, with $y=0$ and $\frac{dy}{dx}=1$ when $x=0$. It is solved numerically with step length $h=0.1$, where $x_n=0.1n$ and $y_n$ approximates $y$ at $x_n$, using $\left(\frac{d^2y}{dx^2}\right)_n=\frac{y_{n+1}-2y_n+y_{n-1}}{h^2}$ and $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_{n-1}}{2h}$.
   Show that $y_{n+1}=1.99y_n-y_{n-1}$.3 marks