Euler's method walks along a slope field in straight steps, using the slope at each point to reach the next.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The rule
The new y is the old y plus the slope times the step size. It is the tangent line approximation, applied repeatedly.
Step size
A smaller step follows the curve more closely and takes more steps. Accuracy and effort trade off directly.
The error accumulates
Each step starts from an already approximate point, so errors compound. The estimate drifts further with each step.
Which way it errs
On a concave-up solution the tangent lies below the curve, so Euler underestimates. Concave down overestimates.
Why it matters
Most differential equations have no closed-form solution. A numerical method is often the only route to an answer.
Following the slope field numerically
Euler's method starts at the initial condition and steps forward along the tangent line, recomputing the slope at each new point. It approximates a solution curve without solving anything.
The step
yₙ₊₁ = yₙ + h·f(xₙ, yₙ), where h is the step size. Each step is a linear approximation, so the method is repeated linearisation — the same idea as tangent line approximation, applied over and over.
Error and step size
Smaller steps give better accuracy but need more of them. The error accumulates, and for a concave-up solution Euler consistently underestimates. Predicting the direction of the error is an exam question.
Show the table
Free-response answers should show each step with its x, y and slope. The exam grades the process as much as the final value, and an unlabelled number earns little.
Step 2: Try It Yourself
Tap and try it out.
- The equationdy/dx = a·y
- Through(0, 1)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: Watch an Example
One step at a time.
Watch Ines Take Two Euler Steps
dy/dx = y with y(0) = 1, using a step size of 0.5.
- Step 1
At (0, 1) the slope is y = 1.
Step 4: Your Turn
Practice makes it stick.
The First Step
Problem 1 of 2
dy/dx = y with y = 1 and a step of 0.5. What is the new y?
The Second Step
Problem 2 of 2
From y = 1.5 with slope 1.5 and step 0.5. What is the new y?
Step Along
1 of 8
y = 2, slope 3, step 0.5. New y?
2 of 8
y = 5, slope 2, step 1. New y?
3 of 8
y = 10, slope −3, step 0.5. New y?
4 of 8
Does a smaller step size improve accuracy? 1 yes, 0 no.
5 of 8
On a concave-up solution, does Euler over- or underestimate? 1 over, 2 under.
6 of 8
Going from x = 0 to x = 2 with step 0.5. How many steps?
7 of 8
Put one Euler step in order.
- 1Multiply that slope by the step size.
- 2Add the result to the current y.
- 3Advance x by the step size and repeat.
- 4Evaluate the slope at the current point.
8 of 8
y = 4, slope 6, step 0.25. New y?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = 3, slope 4, step 0.5. What is the new y?
Question 2 of 2
Why does Euler underestimate a concave-up solution?
What You Learned
- Euler's method steps along tangent lines through a slope field.
- New y is old y plus slope times step size.
- Concave up underestimates and concave down overestimates.