Numerical solution of differential equationsEdexcel A-Level Further Maths: Subtopic test

1. 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 $x_n$.

   (a)

   Find the approximation $y_1$ to $y(0.2)$.

   [1 mark]

   • A$1.24$
   • B$1.2$
   • C$1.1$
   • D$2$

   (b)

   Find the approximation $y_2$ to $y(0.4)$.

   [1 mark]

   • A$1.44$
   • B$1.52$
   • C$1.4$
   • D$1.48$

   (c)

   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]

   Total for question 1: 4 marks

2. 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 $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_n}{h}$.

   (a)

   Which formula gives $y_{n+1}$ in terms of earlier values?

   [1 mark]

   • A$y_{n+1}=y_{n-1}+2h(x_n-y_n)$
   • B$y_{n+1}=y_n+2h(x_n-y_n)$
   • C$y_{n+1}=y_{n-1}+h(x_n-y_n)$
   • D$y_{n+1}=y_n+h(x_{n+1}-y_{n+1})$

   (b)

   Find $y_2$.

   [1 mark]

   • A$0.82$
   • B$0.86$
   • C$0.84$
   • D$1.2$

   (c)

   Find $y_3$.

   [2 marks]

   Total for question 2: 4 marks

3. 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 $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_{n-1}}{2h}$.

   (a)

   Show that $y_{n+1}=1.99y_n-y_{n-1}$.

   [3 marks]

   (b)

   Use the formula in part (a) with $n=0$, together with the approximation for $\frac{dy}{dx}$, to find $y_1$. Hence find $y_2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $y$ satisfies $\frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=0$, with $y=1$ and $\frac{dy}{dx}=0$ when $x=0$. It is solved numerically with $h=0.5$, where $x_n=0.5n$ 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 $\left(\frac{dy}{dx}\right)_n=\frac{y_{n+1}-y_{n-1}}{2h}$.

   (a)

   (i) Show that $6y_{n+1}-7y_n+2y_{n-1}=0$.
   (ii) Find $y_1$.

   [6 marks]

   (b)

   (i) Find $y_2$, an approximation to $y(1)$.
   (ii) The exact solution is $y=(1+x)e^{-x}$. Calculate the percentage error in $y_2$.
   (iii) State, with a reason, how the approximation could be improved.

   [6 marks]

   Total for question 4: 12 marks

End of questions