91Ó°ÊÓ

Let's dive into the world of first-order linear differential equations. These equations can feel complex, but they follow a structured form that allows us to solve them systematically. A first-order linear differential equation has the general format:

• \( y' + P(x)y = Q(x) \)

- **\( y' \):** This represents the derivative of \( y \) with respect to \( x \).- **\( P(x) \):** A given function of \( x \) that appears as a coefficient of \( y \).- **\( Q(x) \):** Another function of \( x \), representing the non-homogeneous part of the differential equation.
The equation \( y' + y \tan x = -\sin x \) clearly fits this form when you match \( P(x) = \tan x \) and \( Q(x) = -\sin x \). Understanding this structure helps us categorize and approach the problem using established techniques.

The integrating factor is a pivotal tool in solving linear differential equations efficiently. It transforms our equation into a form that is easy to solve by integration.
To find the integrating factor, we use the formula:

• \( \mu(x) = e^{\int P(x)\, dx} \)

This formula involves calculating an exponential function with the integral of \( P(x) \). In our example with \( P(x) = \tan x \), integrating gives us:

• \( \int \tan x \, dx = -\ln |\cos x| \)

Thus, the integrating factor becomes:

• \( \mu(x) = e^{ - \ln |\cos x| } = \sec x \)

By applying this integrating factor, we simplify the differential equation, making it much more "friendly" to solve.

Once we have the integrating factor, the next step is solving the modified equation. This process involves:

• Multiplying the entire original differential equation by the integrating factor \( \mu(x) = \sec x \).
• This yields the equation:\[ (y \sec x)' = -\sin x \sec x \]

This new equation is much more manageable. The left-hand side is the derivative of a product, making it easy to integrate.

• Integrating both sides results in:\[ y \sec x = \ln |\cos x| + C \]

Here, \( C \) is the constant of integration, accounting for any potential constants arising from a given specific condition.

Integration is vital in solving differential equations, but it can sometimes feel tricky. Thankfully, we have a suite of techniques to apply depending on the function we are working with.
In the solution process for our differential equation, we had to integrate \( -\tan x \). This antiderivative:

• \[ \int - \tan x \, dx = \ln |\cos x| \]

illustrates the use of a basic substitution method or recognizing derivatives of standard functions. Often, recognizing the form of functions we know how to integrate can save significant time.

• Think about where substitution, integration by parts, or partial fractions might simplify a problem.
• Sometimes rewriting a function into a form that is recognizable can be done by factoring, expanding, or manipulating trigonometric identities.

These techniques boost your ability to tackle more complex integrals in differential equations. As you practice, you'll become more adept at deciding which method to apply and when.