first order linear differential equation

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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, from physics and engineering to biology and economics. Understanding how to solve them is crucial for many applications. This article will guide you through the definition, solution methods, and examples of 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 first 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). They can't involve y or its derivatives.

The key characteristic is that the equation is linear in y and its derivative. This means that y and dy/dx appear only to the first power and are not multiplied together.

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, denoted by μ(x), which is chosen to make the left-hand side a perfect derivative.

1. Finding the Integrating Factor:

The integrating factor is given by:

μ(x) = e^(∫P(x)dx)

2. Multiplying the Equation:

Multiply the original differential equation by the integrating factor μ(x):

μ(x)dy/dx + μ(x)P(x)y = μ(x)Q(x)

3. Recognizing the Perfect Derivative:

The left-hand side of this equation is now the derivative of the product μ(x)y:

d/dx[μ(x)y] = μ(x)Q(x)

4. Integrating Both Sides:

Integrate both sides of the equation with respect to x:

∫d/dx[μ(x)y] dx = ∫μ(x)Q(x) dx

This simplifies to:

μ(x)y = ∫μ(x)Q(x) dx + C

where C is the constant of integration.

5. Solving for y:

Finally, solve for y by dividing by μ(x):

y = [∫μ(x)Q(x) dx + C] / μ(x)

Examples of First-Order Linear Differential Equations

Let's illustrate the method with a couple of examples.

Example 1:

Solve the differential equation: dy/dx + 2xy = x

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

2. Find the integrating factor: μ(x) = e^(∫2x dx) = e^(x²)

3. Multiply and integrate:

   e^(x²) dy/dx + 2xe^(x²)y = xe^(x²)

   d/dx[ye^(x²)] = xe^(x²)

   ∫d/dx[ye^(x²)] dx = ∫xe^(x²) dx

   ye^(x²) = (1/2)e^(x²) + C

4. Solve for y: y = 1/2 + Ce^(-x²)

Example 2: A more complex example

Solve the differential equation: dy/dx + (1/x)y = x² for x>0

1. Identify P(x) and Q(x): P(x) = 1/x, Q(x) = x²

2. Find the integrating factor: μ(x) = e^(∫(1/x) dx) = e^(ln|x|) = x (since x>0)

3. Multiply and integrate:

   x dy/dx + y = x³

   d/dx(xy) = x³

   ∫d/dx(xy) dx = ∫x³ dx

   xy = (1/4)x⁴ + C

4. Solve for y: y = (1/4)x³ + C/x

Applications of First-Order Linear Differential Equations

First-order linear differential equations have a wide range of applications:

• Growth and Decay: Modeling population growth, radioactive decay, or compound interest.
• Circuit Analysis: Analyzing electrical circuits with resistors, capacitors, and inductors.
• Mixing Problems: Determining the concentration of a substance in a tank over time.
• Newton's Law of Cooling: Calculating the temperature of an object as it cools.

Understanding and mastering the solution techniques for these equations is essential for success in many scientific and engineering disciplines. Remember to always check your solutions by substituting them back into the original differential equation.