What does Euler's method do geometrically?

Follows the tangent at the start of each step

State the improved Euler method.

y{r+1}=y{r-1}+2hf(xr,yr), x{r+1}=xr+h

How do you get y1 in the improved method?

Use ordinary Euler for the first step

Why is the improved method better?

It uses the gradient at the midpoint of the interval.

What does a smaller h do?

Gives a more accurate answer but needs more steps.

When does Euler's method underestimate?

When the solution curve is convex, y''0

Percentage error formula?

\frac{|estimate-exact|}{exact}\times100

Common Euler mistake with x and y?

Using x{r+1} or y{r+1} in the gradient instead of xr,yr

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

Card 1 of 120 of 12 known

Question

State Euler's method.

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

State Euler's method.: $y_{r+1}=y_r+hf(x_r,y_r)$ , $x_{r+1}=x_r+h$
What does Euler's method do geometrically?: Follows the tangent at the start of each step
What is $h$ ?: The step length in $x$
State the improved Euler method.: $y_{r+1}=y_{r-1}+2hf(x_r,y_r)$ , $x_{r+1}=x_r+h$
How do you get $y_1$ in the improved method?: Use ordinary Euler for the first step
Why is the improved method better?: It uses the gradient at the midpoint of the interval.
What does a smaller $h$ do?: Gives a more accurate answer but needs more steps.
When does Euler's method underestimate?: When the solution curve is convex, $y''>0$
Euler step for $y'=x+y$ , $(0,1)$ , $h=0.1$ ?: $y_1=1+0.1(0+1)=1.1$
Improved step: $y'=x+y$ , $y_0=1$ , $y_1=1.1$ , $h=0.1$ ?: $y_2=1+0.2(1.2)=1.24$
Percentage error formula?: $\frac{|\text{estimate}-\text{exact}|}{\text{exact}}\times100$
Common Euler mistake with $x$ and $y$ ?: Using $x_{r+1}$ or $y_{r+1}$ in the gradient instead of $x_r,y_r$

Exam questions on Euler's methods for differential equations

A curve satisfies $\frac{dy}{dx}=x+y$ with $y=1$ when $x=0$ . Use a step length $h=0.1$ .
Using $y_1=1.1$ from Euler's method, use the improved Euler method to estimate $y$ when $x=0.2$ .2 marks
A curve satisfies $\frac{dy}{dx}=x^2-y$ with $y=2$ when $x=1$ . Use Euler's method with step length $h=0.2$ .
Explain why a smaller value of $h$ usually gives a more accurate estimate, and state the cost of using it.2 marks
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.
Use Euler's method to estimate $T$ when $t=2$ and when $t=4$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).