Separation of Variables Calculator

Exponential Growth/Decay Calculator (Solved via Separation of Variables)

Initial Quantity (y₀):

Growth/Decay Rate (k):

Time Elapsed (t):

Result:

Final Quantity (y(t)):

function calculateExponentialGrowth() { var initialValue = parseFloat(document.getElementById('initialValue').value); var rateConstant = parseFloat(document.getElementById('rateConstant').value); var timeValue = parseFloat(document.getElementById('timeValue').value); if (isNaN(initialValue) || isNaN(rateConstant) || isNaN(timeValue)) { document.getElementById('finalValue').textContent = 'Please enter valid numbers for all fields.'; return; } var finalQuantity = initialValue * Math.exp(rateConstant * timeValue); document.getElementById('finalValue').textContent = finalQuantity.toFixed(4); }

Understanding Separation of Variables for Exponential Models

The method of separation of variables is a powerful technique used to solve certain types of first-order ordinary differential equations. It is particularly effective when the differential equation can be rearranged so that all terms involving the dependent variable (and its differential) are on one side of the equation, and all terms involving the independent variable (and its differential) are on the other side.

The General Idea

Consider a differential equation of the form dy/dx = f(x)g(y). To solve this using separation of variables, we first separate the variables:

dy/g(y) = f(x)dx

Then, we integrate both sides:

∫ (1/g(y)) dy = ∫ f(x) dx

Solving these integrals and rearranging the resulting equation gives the general solution to the differential equation.

Application: Exponential Growth and Decay

One of the most common and illustrative applications of separation of variables is in solving differential equations that model exponential growth or decay. These are typically represented by the equation:

dy/dt = k * y

Where:

y is the quantity that is changing.
t is time.
k is the constant of proportionality (growth rate if k > 0, decay rate if k < 0).

Solving `dy/dt = k * y` using Separation of Variables

Separate the variables:
Divide both sides by y and multiply by dt:

(1/y) dy = k dt
Integrate both sides:
∫ (1/y) dy = ∫ k dt

This yields:

ln|y| = kt + C (where C is the constant of integration)
Solve for y:
Exponentiate both sides:

|y| = e^(kt + C)

|y| = e^(kt) * e^C

Let A = ±e^C (or just A = e^C if we consider y to be positive, which is common in growth/decay models). Then:

y(t) = A * e^(kt)
Determine the constant A:
If we know the initial quantity at time t=0, let's call it y₀, then:

y₀ = A * e^(k*0)

y₀ = A * e^0

y₀ = A * 1

So, A = y₀.

Thus, the particular solution for exponential growth/decay is:

y(t) = y₀ * e^(kt)

How to Use the Calculator

This calculator helps you find the final quantity y(t) for an exponential growth or decay model, given its initial quantity, rate constant, and the time elapsed. Simply input the values into the fields:

Initial Quantity (y₀): The starting amount or value at t=0.
Growth/Decay Rate (k): The constant rate at which the quantity grows (positive k) or decays (negative k). This is often expressed as a decimal (e.g., 0.05 for 5% growth).
Time Elapsed (t): The duration over which the growth or decay occurs.

Example Calculation

Suppose you have an initial population of 100 bacteria that grows at a continuous rate of 5% per hour. You want to know the population after 10 hours.

Initial Quantity (y₀) = 100
Growth Rate (k) = 0.05
Time Elapsed (t) = 10

Using the formula y(t) = y₀ * e^(kt):

y(10) = 100 * e^(0.05 * 10)

y(10) = 100 * e^(0.5)

y(10) ≈ 100 * 1.64872

y(10) ≈ 164.872

The calculator will show that the final quantity is approximately 164.87.