Diff Eq Calculator

Exponential Growth/Decay Differential Equation Calculator

This calculator solves the first-order linear differential equation of the form dy/dt = k * y, which models exponential growth or decay. The solution to this differential equation is given by the formula y(t) = y₀ * e^(k*t), where y(t) is the value at time t, y₀ is the initial value, k is the growth/decay rate, and e is Euler's number (approximately 2.71828).

function calculateExponential() { var initialValue = parseFloat(document.getElementById('initialValue').value); var growthDecayRate = parseFloat(document.getElementById('growthDecayRate').value); var timeValue = parseFloat(document.getElementById('timeValue').value); var resultDiv = document.getElementById('result'); if (isNaN(initialValue) || isNaN(growthDecayRate) || isNaN(timeValue)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } var finalValue = initialValue * Math.exp(growthDecayRate * timeValue); resultDiv.innerHTML = "The value at time t (y(t)) is: " + finalValue.toFixed(4) + ""; }

Understanding Exponential Growth and Decay

Differential equations are fundamental tools in mathematics and science for describing how quantities change. One of the most common and widely applicable differential equations is the first-order linear equation that models exponential growth or decay. This equation is expressed as dy/dt = k * y.

What does dy/dt = k * y mean?

This equation states that the rate of change of a quantity y with respect to time t (dy/dt) is directly proportional to the quantity y itself. The constant of proportionality is k.

• If k > 0, the quantity is growing exponentially (e.g., population growth, compound interest).
• If k < 0, the quantity is decaying exponentially (e.g., radioactive decay, drug concentration in the bloodstream).
• If k = 0, the quantity remains constant.

The Solution: y(t) = y₀ * e^(k*t)

The general solution to the differential equation dy/dt = k * y is y(t) = y₀ * e^(k*t). Let's break down its components:

• y(t): The value of the quantity at a specific time t. This is what the calculator determines.
• y₀ (Initial Value): The value of the quantity at the starting time, usually t=0. This is your baseline.
• e: Euler's number, an irrational mathematical constant approximately equal to 2.71828. It's the base of the natural logarithm.
• k (Growth/Decay Rate): The constant rate at which the quantity is changing. It's crucial to use the correct sign: positive for growth, negative for decay.
• t (Time): The elapsed time from the initial point. The units of t must be consistent with the units of k (e.g., if k is per year, t should be in years).

Practical Applications

This model is incredibly versatile and appears in many fields:

• Biology: Modeling population growth of bacteria, animals, or humans.
• Physics: Describing radioactive decay of isotopes, cooling of objects (Newton's Law of Cooling, which can be reduced to this form under certain assumptions).
• Chemistry: Reaction kinetics, where the rate of a reaction depends on the concentration of reactants.
• Finance: Continuous compound interest (though we avoid financial terms here, the underlying math is identical).

How to Use This Calculator

To use the Exponential Growth/Decay Differential Equation Calculator, simply input the following values:

1. Initial Value (y₀): Enter the starting amount or quantity. For example, if you start with 100 units of a substance.
2. Growth/Decay Rate (k): Input the constant rate. For a 5% growth rate, enter 0.05. For a 2% decay rate, enter -0.02.
3. Time (t): Specify the time point at which you want to find the value. For instance, if you want to know the value after 10 years, enter 10.

Click "Calculate y(t)", and the calculator will provide the final value of the quantity at the specified time t.

Example Calculation:

Let's say you have an initial population of 100 bacteria (y₀ = 100) that grows at a rate of 5% per hour (k = 0.05). You want to know the population after 10 hours (t = 10).

Using the formula y(t) = y₀ * e^(k*t):

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

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

y(10) = 100 * 1.64872 (approximately)

y(10) = 164.872

So, after 10 hours, the population would be approximately 164.87 bacteria.