Euler's Method Calculator

Euler’s Method Calculator

Use this Euler’s method calculator to solve the first-order differential equation with the given initial condition using the Euler’s method. It also offers a step-by-step solution that shows how Euler's (iterative) procedure approximates the solution to the differential equation to find the next point on the solution curve. 

What is Euler’s Method?

“The Euler method is a first-order numerical method used to solve ordinary differential equations (ODEs) with specific initial values”

This method was invented by the Swiss mathematician Leonhard Euler. Basically, Euler's method makes use of the derivatives at a specific point to approximate the function's value at the next point. By using the tangent line, this estimates the solution of the differential equations. 

Hence, it is important to consider that Euler’s method is a simplification of the iterative method and may not be well estimated. So using the smaller step size generally leads to more precise approximations.

Euler’s Method Formula:

y_(n+1) = y_n + h . f(x_n, y_n)

Within an equation: 

• y_n = Current value of a point on the solution
• y_{n + 1} = Approximate value of the solution at the next step (n+1)
• h = Step size, which controls the increment in the independent variable
• f(x_n, y_n) = Function defining the differential equation. It represents the rate of change of the solution (y) at a specific point (x_n, y_n)

Example:

Using the Euler's method with a step size of 1 to approximate the value of x(4) for the initial value problem by having:

• Differential Equation = x'(t) = x(t)
• Initial Condition = x(0) = 1

Solution (Step-by-Step):

Step # 1 - Set Up Initial Values

• Initial time (t0) = 0
• Initial value of x = x0 = 1

Step # 2 - Use the Euler’s Method Formula

An Euler’s equation has different components - get the given values and find the missing ones. Once you have done this, put the values into the formula to approximate the solution of x (4).

Step # 3 - Perform Iterations

We will repeatedly apply the formula four times (n = 0, 1, 2, 3) to approximate x(4). For the user’s convenience, we have engaged these calculations in the form of the table below:

Step # 4 - Interpretation

The approximated value of x(4) is 16. It is calculated by using Euler’s method with a step size of 1 and four iterations. This iteration procedure can be automated with the help of Euler’s method calculator considering the initial value for ODE.