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Euler's Method Calculator

Enter the first-order differential equation, related values, and let this calculator solve it using Euler's Method.

Enter a Function y′=f(x,y)` or `y′=f(t,y)=:

Divide by Using:

Step Size (h):

Initial Time (t₀)

Value (Y₀):

Answer₁

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Euler’s Method Calculator

Use this Euler’s Method Calculator to numerically solve first-order differential equations with a given initial condition. It provides a step-by-step guide showing how Euler’s iterative procedure approximates the solution at successive points along the solution curve.

What is Euler’s Method?

Euler’s method is a first-order numerical technique for approximating solutions to ordinary differential equations (ODEs) given an initial value.

Developed by the Swiss mathematician Leonhard Euler, this approach uses the slope at a known point to estimate the value of the function at the next point. In essence, it follows the tangent line to predict the next value of the solution.

Tip: Smaller step sizes usually yield more accurate approximations.

Euler method

Euler’s Method Formula

y_n+1 = y_n + h \cdot f(x_n, y_n)

y_n: Current solution value at step n
y_n+1: Approximate solution value at the next step (n+1)
h: Step size (increment along the independent variable)
f(x_n, y_n): Differential equation function representing the rate of change at (x_n, y_n)

Example Problem

Approximate x(4) using Euler’s method with a step size h = 1 for the initial value problem:

Differential equation: x'(t) = x(t)
Initial condition: x(0) = 1

Step-by-Step Solution

Step 1: Initialize Values

t₀ = 0 (initial time)
x₀ = 1 (initial solution value)

Step 2: Apply Euler’s Formula

Use the formula to calculate the next point:

x_n+1 = x_n + h \cdot f(t_n, x_n)

Step 3: Iterate

Perform four iterations (n = 0 to 3) to estimate x(4):

Iteration (n)	t_n	x_n	f(t_n, x_n)	x_n+1
0	0	1	f(0,1) = 1	1 + 1×1 = 2
1	1	2	f(1,2) = 2	2 + 1×2 = 4
2	2	4	f(2,4) = 4	4 + 1×4 = 8
3	3	8	f(3,8) = 8	8 + 1×8 = 16

Step 4: Result

The estimated value of x(4) using Euler’s method with step size h = 1 is 16. Reducing the step size or using an automated calculator can improve accuracy.

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