Euler Method ODE Solver

Euler method formula for first-order ODEs

We consider an initial value problem (IVP) of the form

y' = f(x, y),   y(x₀) = y₀.

Choose a step size h. Euler’s method constructs a sequence of approximate values y₁, y₂, … at points x₁, x₂, … using

• xₘ₊₁ = xₘ + h
• yₘ₊₁ = yₘ + h · f(xₘ, yₘ)

In words: at each step you move forward by h in x and update y using the slope given by f(x, y) at the current point.

Euler update in MathML

The update formula can also be written in MathML as

This is the exact computation the calculator performs at each step.

How to use the Euler method calculator

1. Enter the derivative function f(x, y).
   • Use standard JavaScript syntax: x, y, Math.sin(x), Math.exp(x), Math.log(x), etc.
   • Write powers as multiplication or library calls (for example, x*x, y*y, or Math.pow(x, 2)).
   • Do not include y' =; enter only the right-hand side, such as x + y or 0.5 * x - 3 * y.
2. Set the initial values.
   • x0: the starting value of the independent variable (for example, 0).
   • y0: the value of the solution at x0 (for example, 1 for the condition y(0) = 1).
3. Choose a step size h.
   • This is the increment in x between rows of the output table.
   • Smaller h usually gives better accuracy but requires more steps to cover the same interval.
4. Specify the number of steps.
   • The calculator will perform this many Euler updates.
   • The final x value will be x0 + h × (steps).
5. Run the computation.
   • Click Compute to generate the Euler table.
   • The output lists rows with step index, x, y, and often the slope f(x, y) used for that step (depending on implementation).

All fields must be filled with valid numbers (except the function field, which expects an expression) for the calculation to succeed.

Interpreting the results

The calculator produces a discrete sequence of points approximating the continuous solution curve y(x). Each row (xₘ, yₘ) represents the Euler estimate after n steps.

• Early rows show how the solution begins to move away from the initial condition.
• Later rows indicate how numerical error accumulates as you move farther from x0.
• If you know the exact solution, you can compare yₘ to the true value y(xₘ) and observe the error.

If you plot the pairs (xₘ, yₘ) on a graph and connect them with straight lines, you obtain the piecewise linear Euler approximation to the true solution curve.

Worked example: y' = x + y, y(0) = 1

Consider the initial value problem

y' = x + y,   y(0) = 1.

The exact solution is

y(x) = 2e^x - x - 1.

Suppose we choose a step size h = 0.1 and want the first 3 Euler steps starting at x0 = 0, y0 = 1. In the calculator you would enter:

• f(x,y): x + y
• x0: 0
• y0: 1
• step size: 0.1
• steps: 3

The Euler updates are then:

• Step 0 (initial): x0 = 0, y0 = 1, slope f(0,1) = 0 + 1 = 1.
• Step 1: x1 = 0 + 0.1 = 0.1, y1 = 1 + 0.1 × 1 = 1.1.
• Step 2: slope f(0.1, 1.1) = 0.1 + 1.1 = 1.2, so x2 = 0.2, y2 = 1.1 + 0.1 × 1.2 = 1.22.
• Step 3: slope f(0.2, 1.22) = 0.2 + 1.22 = 1.42, so x3 = 0.3, y3 = 1.22 + 0.1 × 1.42 = 1.362.

A portion of the output table produced by the calculator would look like:

If you compute the exact solution at x = 0.3, you get y(0.3) = 2e^{0.3} - 0.3 - 1, which is slightly different from 1.362. The difference illustrates the accumulated numerical error after three Euler steps.

How Euler compares to other ODE methods

Euler’s method is the simplest member of a large family of numerical integrators. It is easy to understand and implement, but it is not very accurate compared with higher‑order methods. The table below summarizes key differences.

This calculator focuses on the classical Euler method because it highlights the basic idea of stepwise integration and slope-based updates. For precise engineering or scientific work, higher‑order methods are normally preferred.

Limitations and assumptions of this calculator

When using the Euler method and this tool, keep the following points in mind:

• First-order ODEs only. The solver handles a single first‑order ODE of the form y' = f(x, y) with one initial condition. Higher‑order equations must be rewritten as systems of first‑order equations, which this simple interface does not support directly.
• Sensitivity to step size. Because Euler is a first‑order method, large step sizes can lead to large errors, oscillations, or unstable behaviour. For better accuracy, reduce h or switch to a higher‑order method in other software.
• Accumulated global error. Even if each step seems locally accurate, the error accumulates as you move further from x0. Long integration intervals with fixed h may deviate significantly from the true solution.
• Poor performance on stiff problems. Stiff ODEs often require implicit or specially designed solvers. Forward Euler with a moderate step size can become unstable or completely wrong for such systems.
• Domain and function restrictions. Your function f(x, y) must be defined and finite for all (x, y) visited during the integration. Expressions that cause division by zero, invalid logarithms, or overflow will lead to NaN or infinite values.
• Educational, not safety‑critical. The tool is ideal for learning and demonstration. It should not be relied on by itself for safety‑critical design, control systems, or high‑stakes engineering decisions.

By understanding these assumptions and limitations, you can use the Euler method calculator effectively as a learning aid and as a first approximation tool, while recognizing when more advanced methods or professional software are needed.