91Ó°ÊÓ

In first-order linear differential equations, the integrating factor is a special function used to make the equation easy to solve. It is particularly useful when your equation looks like this:

• \( y' + P(x)y = Q(x) \)

The integrating factor, often denoted by \( \mu(x) \), helps us to turn this equation into a form that can easily be integrated. To find it, we use:

• \( \mu(x) = e^{\int P(x) \, dx} \)

In our specific problem, \( P(x) = \tan x \). Evaluating the integral \( \int \tan x \, dx \), we find it equals \( \ln |\sec x| \). Thus, \( \mu(x) = |\sec x| \).
Since the absolute value does not affect the interval of the solution, \( \mu(x) = \sec x \) becomes our integrating factor. This crucial step transforms the differential equation into a solvable form.

An initial condition is a requirement that specifies the value of the solution at a particular point. It ensures the solution to the differential equation is unique. For example, in our problem, the initial condition given is:

• \( y(0) = -1 \)

This means that when \( x = 0 \), the value of \( y \) should be \(-1\). Initial conditions provide a starting point which we can use to determine any constants of integration that appear in our solution.
In this exercise, after finding the general solution of the original differential equation, we substitute \( x = 0 \) and \( y = -1 \) back into it. This allows us to solve for the constant \( C \), ensuring our solution satisfies the given initial condition.

Solving a first-order linear differential equation typically involves several key steps. Here's an overview of the method used in this example:

• Identify the structure: Ensure the equation is in the form \( y' + P(x)y = Q(x) \).
• Calculate the integrating factor \( \mu(x) \): This turns the equation into a product rule form.
• Multiply every term by the integrating factor to transform the equation.
• Recognize the transformed left side as a derivative of a product \( \frac{d}{dx}(y \cdot \mu(x)) \).
• Integrate both sides of this equation for a complete solution.
• Apply the initial condition to find any unknown constants.

By using these steps, the complex differential equation can be simplified, solved, and adjusted with initial conditions for a specific solution.

Simplifying a differential equation is about reformulating it into a simpler form that reveals its solution more readily. The original equation:

• \( y' + \tan x \cdot y = \cos^2 x \)

becomes much easier after multiplication by the integrating factor \( \sec x \):

• \( \sec x \cdot y' + (-\sec x \cdot \tan x) \cdot y = \cos x \)

This simplification allows identification of the left-hand side as the derivative of a product, specifically \( \frac{d}{dx}(y \cdot \sec x) = \cos x \). Recognizing the product derivative simplifies the solution process remarkably. Further simplification involves integration, followed by rearranging to solve for \( y \).
This systematic simplification takes seemingly complicated equations and turns them into more manageable expressions, making the solving process much more straightforward.