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Euler’s method estimates the solution of a differential equation \(\frac{dy}{dx} = f(x, y)\) by taking small straight steps along the slope: \(y_{n+1} = y_n + h\,f(x_n, y_n)\) and \(x_{n+1} = x_n + h\). A smaller step size \(h\) usually gives a more accurate answer.
How to Use the Euler’s Method Calculator
- Type the right side of \(\frac{dy}{dx} = f(x, y)\), such as
x + y. - Enter the starting point \(x_0\) and \(y_0\), the step size \(h\), and the number of steps.
- Press Solve. You get the final estimate, a full table of every step, and a graph of the approximate solution.
How Euler’s Method Works
A differential equation tells you the slope of the solution at every point. Euler’s idea is simple:
- Start at the known point \((x_0, y_0)\).
- Find the slope there: \(f(x_0, y_0)\).
- Walk a short distance \(h\) along that slope to reach a new point.
- Repeat from the new point.
Each step uses the tangent line, just like linear approximation.
Worked Example: \(y’ = x + y\), \(y(0) = 1\), \(h = 0.1\)
| n | \(x_n\) | \(y_n\) | slope \(x_n + y_n\) | \(y_{n+1}\) |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1.1 |
| 1 | 0.1 | 1.1 | 1.2 | 1.22 |
| 2 | 0.2 | 1.22 | 1.42 | 1.362 |
| 3 | 0.3 | 1.362 | 1.662 | 1.5282 |
After 10 steps, Euler’s method gives \(y(1) \approx 3.1875\). The exact solution is \(y = 2e^x – x – 1\), so \(y(1) = 2e – 2 \approx 3.4366\). The error is about 0.25. With \(h = 0.01\) the error drops to about 0.027.
Accuracy and Step Size
- Euler’s method is a first-order method: cutting \(h\) in half roughly cuts the error in half.
- If the solution curves up, Euler’s method tends to underestimate, because each tangent line sits below the curve.
- More accurate methods, like the improved Euler method and Runge–Kutta (RK4), use more than one slope per step.
Common Mistakes
- Using the new x with the old y. The slope must use \(x_n\) and \(y_n\) from the same step.
- Forgetting to multiply by h. The change in \(y\) is \(h \times\) slope, not just the slope.
- Rounding too early. Keep several decimal places until the final answer.
Learn the Math Behind the Euler's Method Calculator
Euler’s method follows tangent lines in small steps to approximate the solution of a differential equation.
Frequently Asked Questions
Why is there no ✓ Verified badge?
Euler’s method gives an estimate, not an exact answer, so there is nothing exact to check against. The table shows every number so you can check each step by hand.
How many steps should I use?
Divide the distance you want to travel by \(h\). To go from \(x = 0\) to \(x = 1\) with \(h = 0.1\), use 10 steps.
Can I use t instead of x?
Yes. Type the equation with t and y. The calculator treats t the same as x.
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