Differential Equation Solver (Euler's Method)
This calculator numerically solves a first-order ordinary differential equation (ODE) of the form dy/dx = f(x, y) using Euler's method. Euler's method is a simple, first-order numerical procedure for solving ODEs with a given initial value.
Solution Steps:
| Step | x | y |
|---|---|---|
| " + j + " | " + results[j].x.toFixed(6) + " | " + results[j].y.toFixed(6) + " |
Understanding Differential Equations and Euler's Method
A differential equation is a mathematical equation that relates a function with its derivatives. In many scientific and engineering fields, these equations are fundamental for describing how quantities change over time or space. For example, they can model population growth, the motion of planets, the flow of heat, or the decay of radioactive materials.
A first-order ordinary differential equation (ODE) typically takes the form dy/dx = f(x, y), where y is an unknown function of x, and f(x, y) is a given function. Solving such an equation means finding the function y(x) that satisfies it. While some differential equations can be solved analytically (giving an exact formula for y(x)), many cannot. In these cases, numerical methods are used to approximate the solution.
What is Euler's Method?
Euler's method is one of the simplest and most intuitive numerical methods for approximating solutions to first-order ODEs with a given initial condition. It works by taking small steps along the tangent line of the solution curve. Given an initial point (x₀, y₀) and the derivative dy/dx = f(x, y), Euler's method estimates the next point (x₁, y₁) using the following formulas:
y_new = y_old + h * f(x_old, y_old)x_new = x_old + h
Here, h is the "step size," a small increment in x. The method essentially approximates the curve with a series of short line segments. The smaller the step size h, the more accurate the approximation generally becomes, but it also requires more computational steps.
How to Use the Calculator
- Function f(x, y): Enter the right-hand side of your differential equation
dy/dx = f(x, y). You can use standard JavaScript math functions (e.g.,Math.sin(x),Math.cos(y),Math.exp(x),Math.log(y),Math.pow(x, 2)for x²). - Initial x (x₀): This is the starting value for
x. - Initial y (y₀): This is the starting value for
y, corresponding tox₀. Together,(x₀, y₀)form the initial condition. - Step Size (h): This determines the increment for
xat each step. Smaller values generally lead to more accurate results but require more steps. - Number of Steps (N): This specifies how many times Euler's method will be applied. The final
xvalue will bex₀ + N * h.
After entering your values, click "Calculate Solution" to see a table of approximated x and y values at each step.
Example Calculation:
Let's solve the differential equation dy/dx = x + y with initial condition y(0) = 1. We'll use a step size h = 0.1 and N = 10 steps.
- Function f(x, y):
x + y - Initial x (x₀):
0 - Initial y (y₀):
1 - Step Size (h):
0.1 - Number of Steps (N):
10
Step 0: x = 0, y = 1 (Initial Condition)
Step 1:
f(0, 1) = 0 + 1 = 1y_new = 1 + 0.1 * 1 = 1.1x_new = 0 + 0.1 = 0.1- Result: (x=0.1, y=1.1)
Step 2:
f(0.1, 1.1) = 0.1 + 1.1 = 1.2y_new = 1.1 + 0.1 * 1.2 = 1.1 + 0.12 = 1.22x_new = 0.1 + 0.1 = 0.2- Result: (x=0.2, y=1.22)
The calculator will continue this process for all 10 steps, providing a table of x and y values, showing the approximate solution curve.