What Euler's Method Is For
Sometimes a differential equation tells you how a quantity is changing, but finding an exact formula for the quantity is difficult or impossible. In those situations, you can still approximate the solution using the information you do have.
Euler's Method is a numerical procedure that uses the slope given by a differential equation to estimate the next point on a solution curve. Starting from a known point, you move a small horizontal distance and use the current slope to predict where the solution will be next.
Think of it as repeatedly asking: "If the solution has this slope right now, where will it approximately be after a small step?"
Known point → find the slope → take a small step → get a new point → repeat.
The Euler's Method Formula
Suppose a differential equation is written as:
and you know a point on the solution:
For a step size , Euler's Method estimates the next point using:
The two equations work together. First, evaluate the differential equation at the current point to find the slope. Then use that slope to estimate the new value.
Why the Formula Works
Recall the point-slope idea from algebra. If you know a point and the slope of a line, you can estimate another point using:
Euler's Method does essentially the same thing. The difference is that the slope can change from point to point because the differential equation gives you a potentially different slope at every location.
At each step, Euler's Method temporarily treats the solution curve as though it were a straight line with the current slope.
Current point:
Current slope:
Horizontal step:
Vertical change:
Next point:
A Step-by-Step Method
- Write down the differential equation. Identify the expression for the derivative, such as .
- Record the starting point and step size. The initial condition gives your first and .
- Evaluate the differential equation at the current point to find the current slope.
- Calculate the change in . Multiply the slope by the step size.
- Find the next . Add the change to the current .
- Repeat until you reach the requested -value.
Worked Examples
Suppose:
Use Euler's Method with step size to approximate .
Consider:
Use Euler's Method with step size to approximate .
Suppose:
Use Euler's Method with step size , starting from .
Euler's Method can also be used to approximate a solution at an earlier -value, provided the problem gives an appropriate step direction.
Suppose:
To move from to , use a negative step size.
Euler's Method as a Recursion
For repeated calculations, it helps to think of Euler's Method as a recursion. If the current approximation is , then the next approximation is:
At the same time, the -coordinate advances by the step size:
So the complete Euler update can be written as:
This is the version worth remembering for AP Calculus. Every new point is created from the previous point and the slope at the previous point.
Euler's Method and Slope Fields
Euler's Method connects directly to slope fields. A slope field shows the direction of the solution curve at many points, while Euler's Method actually uses those slopes to construct an approximate path.
If you begin at a known point and repeatedly follow the small line segments indicated by the differential equation, you are essentially performing Euler's Method.
Slope field: Shows what direction the solution should travel at each point.
Euler's Method: Uses those directions numerically to estimate where the solution travels.
Exact solution: The actual curve that Euler's Method is trying to approximate.
Accuracy and Step Size
Euler's Method is an approximation, so the result will generally not be exactly equal to the true solution. One of the most important ideas is that the step size affects the accuracy.
A smaller step size usually gives a better approximation because the solution curve has less opportunity to bend away from the straight-line approximation during each individual step.
For example, compare using:
With the smaller step size, Euler's Method takes more steps, but each step is shorter. This generally reduces the approximation error.
Do not confuse the number of steps with the size of each step. A problem may ask for several Euler steps with a small , or only a few steps with a larger one.
Common Mistakes
The slope for an Euler step is evaluated at the current point. Find the slope first, then use it to calculate the next point.
The vertical change is not simply the slope. It is . The step size tells you how far you move horizontally.
After each Euler step, the new point becomes the starting point for the next step. You must update both the and values.
If you are moving toward smaller -values, the step size is negative. A sign error can change every subsequent approximation.
If a calculator is allowed, keep several decimal places during intermediate calculations and round only the final answer. Early rounding can accumulate over multiple Euler steps.
Practice Problems
The differential equation has the initial condition . Use Euler's Method with step size to approximate .
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The differential equation has . Use Euler's Method with step size to approximate .
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Given , use Euler's Method with step size to approximate .
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Given , use Euler's Method with step size to approximate . Round your answer to three decimal places.
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What to Remember for the AP Exam
When you see an Euler's Method problem, don't overthink it. The process is mechanical once you organize the information correctly.
1. Start with the given point.
2. Evaluate the differential equation to get the current slope.
3. Multiply the slope by the step size.
4. Add that change to the current .
5. Update by the step size.
6. Repeat until you reach the requested point.
The key formula is:
Once you understand that formula, Euler's Method becomes a repeated application of the same two calculations.
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