first order linear differential equation

Natasha willow

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19-03-2025

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First-order linear differential equations are a fundamental type of differential equation encountered in various fields like physics, engineering, and economics. Understanding how to solve them is crucial for many applications. This article provides a comprehensive guide to understanding and solving these equations.

What is a First-Order Linear Differential Equation?

A first-order linear differential equation is an equation that can be written in the standard form:

dy/dx + P(x)y = Q(x)

where:

• dy/dx represents the derivative of the dependent variable y with respect to the independent variable x.
• P(x) and Q(x) are functions of x only. They can be constants, but more often they are functions of x.

This form is crucial because it allows us to use a specific method for solving the equation. Note that if the equation isn't in this form, you might need to manipulate it algebraically to get it there.

Solving First-Order Linear Differential Equations: The Integrating Factor Method

The most common method for solving first-order linear differential equations is the integrating factor method. This involves multiplying the entire equation by an integrating factor, I(x), which is chosen to make the left-hand side a perfect derivative.

The integrating factor is defined as:

I(x) = e^∫P(x)dx

Notice that the integral of P(x) is just any antiderivative – we don't need the constant of integration here.

Here's the step-by-step process:

1. Identify P(x) and Q(x): Rewrite your equation in the standard form (dy/dx + P(x)y = Q(x)) to clearly identify P(x) and Q(x).

2. Calculate the integrating factor: Compute the integral of P(x) and then find the exponential of this integral.

3. Multiply the equation by the integrating factor: Multiply both sides of the original differential equation by I(x).

4. Recognize the perfect derivative: The left side of the equation, after multiplication by the integrating factor, will be the derivative of the product I(x)y with respect to x. This can be written as d/dx[I(x)y].

5. Integrate both sides: Integrate both sides of the equation with respect to x. The left side will simplify nicely due to the previous step.

6. Solve for y: Solve the resulting equation for y to obtain the general solution. Remember to include the constant of integration.

7. Apply initial conditions (if provided): If an initial condition (a specific value of y at a specific value of x) is given, substitute it into the general solution to find the particular solution.

Example: Solving a First-Order Linear Differential Equation

Let's solve the differential equation:

dy/dx + 2xy = x

1. Identify P(x) and Q(x): Here, P(x) = 2x and Q(x) = x.

2. Calculate the integrating factor: ∫P(x)dx = ∫2x dx = x² I(x) = e^x²

3. Multiply by the integrating factor: e^x²(dy/dx + 2xy) = xe^x²

4. Recognize the perfect derivative: The left side is d/dx[ye^x²].

5. Integrate both sides: ∫d/dx[ye^x²] dx = ∫xe^x² dx ye^x² = (1/2)e^x² + C (where C is the constant of integration)

6. Solve for y: y = (1/2) + Ce^-x²

This is the general solution to the differential equation. If an initial condition were given (e.g., y(0) = 1), we could substitute it to find the specific value of C and thus the particular solution.

Applications of First-Order Linear Differential Equations

First-order linear differential equations appear in numerous applications:

• Newton's Law of Cooling: Describes the temperature change of an object as it cools or heats in an environment.

• Radioactive Decay: Models the decay of a radioactive substance over time.

• RL Circuits: Analyzes the current flow in an electrical circuit with a resistor and an inductor.

• Population Growth (with limitations): Describes the growth of a population subject to factors like resource limitations (Logistic growth).

Understanding and mastering the solution techniques for first-order linear differential equations is essential for anyone working in fields involving mathematical modeling and problem-solving. The integrating factor method provides a powerful and systematic way to find solutions, applicable across a wide range of practical situations.