Euler's methods for differential equationsAQA A-Level Further Maths: Subtopic test

1. 1
   A curve satisfies $\frac{dy}{dx}=x+y$ with $y=1$ when $x=0$. Use a step length $h=0.1$.

   (a)

   Use Euler's method to find $y_1$, the estimate of $y$ when $x=0.1$.

   [1 mark]

   • A$1.1$
   • B$1.11$
   • C$2$
   • D$0.1$

   (b)

   Continue Euler's method to find $y_2$, the estimate of $y$ when $x=0.2$.

   [1 mark]

   • A$1.2$
   • B$1.22$
   • C$1.21$
   • D$2.3$

   (c)

   Using $y_1=1.1$ from Euler's method, use the improved Euler method to estimate $y$ when $x=0.2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve satisfies $\frac{dy}{dx}=x^2-y$ with $y=2$ when $x=1$. Use Euler's method with step length $h=0.2$.

   (a)

   Find the estimate of $y$ when $x=1.2$.

   [1 mark]

   • A$2.2$
   • B$1.0$
   • C$1.8$
   • D$1.9$

   (b)

   Continue to find the estimate of $y$ when $x=1.4$.

   [1 mark]

   • A$1.8$
   • B$1.872$
   • C$1.6$
   • D$1.728$

   (c)

   Explain why a smaller value of $h$ usually gives a more accurate estimate, and state the cost of using it.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cooling object has temperature $T$ °C at time $t$ minutes, where $\frac{dT}{dt}=-0.1(T-20)$ and $T=80$ when $t=0$. Euler's method with step length $h=2$ minutes is used.

   (a)

   Use Euler's method to estimate $T$ when $t=2$ and when $t=4$.

   [3 marks]

   (b)

   The exact solution is $T=20+60e^{-0.1t}$. Find the error in your estimate of $T(4)$, and explain why Euler's method gives an underestimate here.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve satisfies $\frac{dy}{dx}=y-x^2$ with $y=1$ when $x=0$. The exact solution is $y=x^2+2x+2-e^{x}$. Use a step length $h=0.25$.

   (a)

   Use Euler's method for the first step, and then the improved Euler method, to estimate $y$ when $x=0.5$.

   [6 marks]

   (b)

   (i) Using $y_2=1.59375$, use the improved Euler method to estimate $y$ when $x=0.75$.
   (ii) Find the percentage error of this estimate using the exact solution.

   [6 marks]

   Total for question 4: 12 marks

End of questions