Softonic

Euler Method Calculator

Step 1 / 3 Euler method approximation

Enter the equation and initial point

For dy/dx = f(x,y), enter only f(x,y). Use * for multiplication and ^ for powers. Supported functions: sin, cos, tan, exp, log, sqrt, abs. Use parentheses for function arguments. Decimal points in the equation use a dot.

Next

Approximate a first-order initial value problem of the form dy/dx = f(x,y), with y given at an initial x. Enter the right-hand side of the equation, the initial point, a positive step size and the target x. The calculator applies forward Euler steps toward the target and shows every intermediate value. The result is a numerical approximation, not an exact solution.

How to use Euler's method

  1. 1

    Enter the differential equation

    For dy/dx = x + y, enter x + y as f(x,y). Then enter the known initial coordinates x and y.

  2. 2

    Set the interval

    Enter a positive step size h and a target x. The calculator can move to larger or smaller x values.

  3. 3

    Read the approximation

    See the estimated y at the target and a table of x, y, slope and actual step size for up to 200 steps.

What the calculator computes

Starting from (x₀, y₀), forward Euler’s method repeats yₙ₊₁ = yₙ + hₙ f(xₙ, yₙ) and xₙ₊₁ = xₙ + hₙ. Each slope is evaluated at the start of its step. A positive step-size input gives the maximum step length; the calculator applies the appropriate sign when the target lies to the left of the initial point. If the remaining distance is shorter than the chosen step size, the last step is shortened to reach the exact target x.

For dy/dx = x + y, y(0) = 1, h = 0.1 and target x = 0.3:

Step x Approximate y Slope used for next step
0 0 1 1
1 0.1 1.1 1.2
2 0.2 1.22 1.42
3 0.3 1.362 End

The table in the calculator places the slope used for an update on the resulting row, labelled “Previous slope”. Thus the slope 1.42 appears beside step 3. The exact solution for this example is y = 2eˣ - x - 1, which gives approximately 1.3997 at x = 0.3. Euler’s 1.362 is lower because it follows the beginning-of-step tangent at each update. A smaller h generally reduces error for a well-behaved problem, but error depends on the equation and interval. The calculator does not estimate an error bound.

Enter explicit multiplication, such as 2*x, x*y or 2*sin(x). The expression accepts x, y, parentheses, +, -, *, /, ^, constants pi and e, and functions sin, cos, tan, exp, log, sqrt, abs. Use a dot for decimal constants inside the expression, such as 0.5*y. For the separate numeric fields, a local decimal mark or an unambiguous dot works; omit grouping separators.

Euler’s method and its recurrence are described in OpenStax Calculus Volume 2. The LibreTexts numerical methods chapter explains how step size affects approximation error.

Frequently Asked Questions

Enter a positive step size. If target x is less than initial x, the calculator automatically steps backward. The table displays the signed step it used.

The final step is shortened to the remaining distance. For example, h = 0.4 from x = 0 to x = 1 produces steps of 0.4, 0.4 and 0.2, ending at x = 1.

No. Forward Euler follows local tangent slopes and accumulates approximation error. The reported value is an estimate at the target x, not an exact symbolic solution or an error guarantee.

The tool handles a first-order equation written explicitly as dy/dx = f(x,y), with one known initial y value. It does not solve higher-order equations, systems, equations requiring implicit solving or equations undefined along the requested path.

The 200-step limit keeps the table usable and prevents a large browser computation. Increase h or choose a closer target if the requested interval needs more steps.

Yes. The CSV download contains the unrounded numerical x and y values, the slope used to reach each row, and that row’s signed step size. The on-screen table rounds values for readability.

The calculation runs in your browser. In the step-by-step flow, the inputs are saved only in this browser tab, while the URL contains a random job ID rather than the equation or values.

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