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Euler's Method Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An Euler's method calculator is a numerical analysis tool used to approximate solutions to first-order ordinary differential equations (ODEs) with a given initial condition. It starts from a known point and steps forward in fixed horizontal increments (h), using the local derivative (slope) to compute the next y-value linearly. The calculator outputs an iteration table showing steps, coordinates, slopes, and next values.

Solve differential equations dy/dx = f(x, y) numerically and generate a step-by-step calculations table.

Quick Answer

Approximate solutions to ordinary differential equations dy/dx = f(x, y) numerically. Enter initial values and step size to generate a step table.

ODE parameters

Supports algebraic variables x and y. · e.g. x + y

e.g. 0

e.g. 1

e.g. 0.1

e.g. 5

Euler's Method Results

Approximate y at x = 0.5

1.72102

Approximation TableStep | x_n | y_n | dy/dx | y_(n+1) -------------------------------------------------- 0 | 0.0000 | 1.0000 | 1.0000 | 1.1000 1 | 0.1000 | 1.1000 | 1.2000 | 1.2200 2 | 0.2000 | 1.2200 | 1.4200 | 1.3620 3 | 0.3000 | 1.3620 | 1.6620 | 1.5282 4 | 0.4000 | 1.5282 | 1.9282 | 1.7210

Euler's Method approximates the solution linearly. Smaller step sizes (h) yield more accurate results but accumulate floating-point error over many steps.

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Examples

dy/dx = x + y, y(0) = 1, h = 0.1, 5 steps

y(0.5) ≈ 1.7210

dy/dx = y − x, y(0) = 2, h = 0.2, 3 steps

y(0.6) ≈ 3.4240

How it works

The calculator starts at the initial condition and increments coordinates iteratively using the local derivative slope at each step.

Iteration Step · y_(n+1) = y_n + h × f(x_n, y_n)

How Euler's method works

Euler's method approximates the solution curve of a differential equation starting from a known initial point (x0, y0). It works by stepping forward in increments of width h:

  1. Evaluate the derivative (slope) at the current point: slope = f(x_n, y_n).
  2. Calculate the vertical step: dy = slope × h.
  3. Add the vertical step to the current y to find the next y: y_(n+1) = y_n + dy.
  4. Step forward horizontally: x_(n+1) = x_n + h.
  5. Repeat for the desired number of steps.

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Frequently asked questions

Euler's method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It approximates the curve of the solution using tangent line segments.

The formula is: y_(n+1) = y_n + h × f(x_n, y_n), where h is the step size, and f(x, y) represents the derivative dy/dx.

The step size is the horizontal distance between consecutive points. A smaller step size generally leads to a more accurate approximation but requires more steps (computations) to reach the target x-value.

No. It is an approximation method. Because it uses tangent lines to approximate curves, it incurs 'truncation error' at every step, which accumulates as you move further from the initial condition.

Euler's method is a first-order method, meaning its error is proportional to the step size h. Runge-Kutta methods (like RK4) are higher-order methods that compute multiple intermediate slopes per step, leading to much higher accuracy for the same step size.