Eulers Method Calculator

How to Use the Euler’s Method Calculator

This Euler’s Method Calculator is a helpful tool designed to approximate solutions for differential equations using initial conditions and a defined step size. The calculator will guide you through entering the required inputs and understanding the results generated. Follow these steps to efficiently use the calculator:

Step 1: Enter the Initial x and y Values

Begin by entering the initial conditions of your problem:

• Initial x value (x₀): Input the initial x-coordinate, where your calculation will start. This field is required and should be a numeric value.
• Initial y value (y₀): Input the initial y-coordinate corresponding to the initial x value. This value is also required and must be numeric.

Step 2: Define the Step Size

You are required to specify the step size (h), which indicates the interval at which the Euler’s method will progress to approximate the solution:

• Step Size (h): Enter a numeric value between 0.000001 and 1. Choose a smaller step size for greater accuracy, and ensure this field is filled as it is vital for calculations.

Step 3: Set the Target x Value

Input the value of x to which you want to compute the solution:

• Target x value: Specify the x-coordinate at which you would like to evaluate the approximate y value via Euler’s method. This field is required.

Step 4: Select the Differential Equation

Choose the type of differential equation representing the problem you want to solve:

• Select Differential Equation: Pick one from the dropdown options. The available equations are:
  • dy/dx = xy
  • dy/dx = x²
  • dy/dx = y²
  • dy/dx = x + y

Step 5: Review the Results

Once all inputs are provided, the calculator will process the information and display the following results:

• Number of Steps: This field shows the number of increments taken to reach the target x value, calculated as the absolute difference between target x and initial x, divided by the step size.
• Final y Value: The approximated y value at the target x. This is calculated by iteratively applying the selected differential equation using Euler’s method.
• Local Truncation Error: Indicates the potential error accumulated at each step, derived from the square of the step size divided by 2.
• Global Truncation Error: An estimation of total error depending on the product of the step size and the exponential of the distance between the target and initial x values.

Review these results to understand the behavior of the differential equation under the given conditions, and adjust your parameters accordingly for more refined approximations if necessary.