First-Order Linear Differential Equation Solver
Understanding First-Order Linear Differential Equations
Differential equations are mathematical equations that relate a function with its derivatives. They are fundamental in science and engineering, used to model phenomena where quantities change over time or space. From population dynamics to radioactive decay, and even the flow of heat, differential equations provide a powerful framework for understanding how systems evolve.
What is a First-Order Linear Differential Equation?
A first-order linear differential equation is one of the simplest yet most widely applicable types. It involves only the first derivative of an unknown function and the function itself, typically in the form:
dy/dt = k * y
Here:
yis the dependent variable (the quantity changing).tis the independent variable (often time).dy/dtrepresents the rate of change ofywith respect tot.kis a constant, known as the rate constant.
This equation states that the rate of change of y is directly proportional to y itself. If k is positive, it describes exponential growth; if k is negative, it describes exponential decay.
The Analytical Solution
The beauty of this specific differential equation is that it has a straightforward analytical solution. By separating variables and integrating, we arrive at the solution:
y(t) = y₀ * e^(k * t)
Where:
y(t)is the value of the function at timet.y₀(y-naught) is the initial value of the function att = 0.eis Euler's number (approximately 2.71828), the base of the natural logarithm.kis the rate constant.tis the time or independent variable.
How to Use the Calculator
Our calculator helps you quickly find the value of y(t) for a given first-order linear differential equation. Simply input the following values:
- Initial Value (y₀): This is the starting amount or condition of the quantity you are modeling. For example, the initial population size, the initial amount of a radioactive substance, or the initial temperature difference.
- Rate Constant (k): This constant determines how quickly the quantity grows or decays. A positive
kindicates growth, while a negativekindicates decay. Its units depend on the units of time. - Time (t): This is the specific point in time at which you want to know the value of
y.
After entering these values, click "Calculate y(t)" to see the result.
Examples of Application
Example 1: Population Growth
Imagine a bacterial colony that starts with 100 cells and grows at a rate of 5% per hour. We want to know the population after 10 hours.
- Initial Value (y₀) = 100 cells
- Rate Constant (k) = 0.05 (5% growth per hour)
- Time (t) = 10 hours
Using the formula: y(10) = 100 * e^(0.05 * 10) = 100 * e^(0.5) ≈ 100 * 1.6487 ≈ 164.87
The calculator would show approximately 164.87 cells.
Example 2: Radioactive Decay
A sample of a radioactive isotope initially has 500 grams. If its decay rate constant is -0.02 per year, how much of the isotope remains after 30 years?
- Initial Value (y₀) = 500 grams
- Rate Constant (k) = -0.02 (2% decay per year)
- Time (t) = 30 years
Using the formula: y(30) = 500 * e^(-0.02 * 30) = 500 * e^(-0.6) ≈ 500 * 0.5488 ≈ 274.40
The calculator would show approximately 274.40 grams.
Example 3: Cooling of an Object (Newton's Law of Cooling – simplified)
An object's temperature difference from its surroundings decreases at a rate proportional to the difference. If the initial temperature difference is 80 degrees Celsius and the rate constant for cooling is -0.15 per minute, what is the temperature difference after 5 minutes?
- Initial Value (y₀) = 80 degrees
- Rate Constant (k) = -0.15 (per minute)
- Time (t) = 5 minutes
Using the formula: y(5) = 80 * e^(-0.15 * 5) = 80 * e^(-0.75) ≈ 80 * 0.4724 ≈ 37.79
The calculator would show approximately 37.79 degrees Celsius.
This calculator provides a quick and easy way to solve for specific points in time for this common type of differential equation, making it a valuable tool for students and professionals alike.