Calculate Differential Equations

Exponential Growth/Decay Differential Equation Calculator

This calculator solves first-order linear homogeneous ordinary differential equations of the form dy/dx = k*y, which models exponential growth or decay. It provides the value of y at a specified target x, given an initial condition.

function calculateDifferentialEquation() { var rateConstant = parseFloat(document.getElementById("rateConstant").value); var initialY = parseFloat(document.getElementById("initialY").value); var initialX = parseFloat(document.getElementById("initialX").value); var targetX = parseFloat(document.getElementById("targetX").value); var resultDiv = document.getElementById("result"); if (isNaN(rateConstant) || isNaN(initialY) || isNaN(initialX) || isNaN(targetX)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } // The analytical solution for dy/dx = k*y is y(x) = y₀ * e^(k * (x – x₀)) var exponent = rateConstant * (targetX – initialX); var finalY = initialY * Math.exp(exponent); resultDiv.innerHTML = "The value of y at x = " + targetX + " is approximately: " + finalY.toFixed(4) + ""; }

Understanding the Exponential Growth/Decay Differential Equation

Differential equations are mathematical equations that relate a function with its derivatives. They are fundamental in modeling various phenomena in science, engineering, economics, and biology, describing how quantities change over time or space.

The Equation: dy/dx = k*y

This calculator focuses on a specific, yet widely applicable, type of first-order linear ordinary differential equation: dy/dx = k*y. Here:

• y represents the quantity that is changing (e.g., population size, amount of a radioactive substance, value of an investment).
• x represents the independent variable, often time (t), but can be any other variable.
• dy/dx (or dy/dt) is the rate of change of y with respect to x.
• k is the "rate constant" or "proportionality constant".

This equation states that the rate of change of y is directly proportional to the current value of y. If k > 0, it describes exponential growth (e.g., population growth, compound interest). If k < 0, it describes exponential decay (e.g., radioactive decay, drug concentration in the bloodstream).

The Analytical Solution

The beauty of this particular differential equation is that it has a straightforward analytical solution. By separating variables and integrating, we arrive at the formula:

y(x) = y₀ * e^(k * (x - x₀))

Where:

• y(x) is the value of y at a given x.
• y₀ (initialY in the calculator) is the initial value of y at x₀.
• e is Euler's number (approximately 2.71828).
• k (rateConstant in the calculator) is the rate constant.
• x (targetX in the calculator) is the target value of the independent variable.
• x₀ (initialX in the calculator) is the initial value of the independent variable.

How to Use the Calculator

1. Rate Constant (k): Enter the constant that determines the rate of growth or decay. A positive value indicates growth, a negative value indicates decay. For example, 0.02 for 2% growth per unit of x, or -0.05 for 5% decay per unit of x.
2. Initial Value of y (y₀): Input the starting amount or quantity of y. This is the value of y when x is at its initial state.
3. Initial Value of x (x₀): Enter the starting point for your independent variable x. Often, this is 0 for initial time.
4. Target Value of x (x_target): Specify the point in time or value of x at which you want to find the corresponding value of y.
5. Click "Calculate y(x_target)" to see the result.

Example Scenario

Imagine a bacterial colony starts with 1000 cells (y₀ = 1000) at time x = 0 (x₀ = 0). If the bacteria grow at a continuous rate of 2% per hour (k = 0.02), what will be the population after 10 hours (x_target = 10)?

• Rate Constant (k): 0.02
• Initial Value of y (y₀): 1000
• Initial Value of x (x₀): 0
• Target Value of x (x_target): 10

Using the formula: y(10) = 1000 * e^(0.02 * (10 - 0)) = 1000 * e^(0.2)

The calculator will show that the population after 10 hours will be approximately 1221.40 cells.