91Ó°ÊÓ

When dealing with first-order linear differential equations, the integrating factor (\( \mu(x) \)) is a crucial tool to simplify and solve these equations. Such equations have the general form:

• \( y' + P(x)y = Q(x) \).

The integrating factor is derived from the function \( P(x) \), which in this case is determined by calculating:

• \( \mu(x) = e^{\int P(x) \, dx} \).

This factor essentially transforms the differential equation into one that can be solved more easily. After finding \( \mu(x) \), you multiply the entire original differential equation by this factor. This process helps rewrite the equation’s left-hand side in a form that can be identified as an exact derivative. The integrating factor serves as a bridge to help organize the equation for integration and eventual solution.

An exact derivative arises when the left side of the differential equation, after being multiplied by the integrating factor, can be rewritten as the derivative of a product of functions. For the current exercise, after applying the factor \( e^{2 \sin x} \), the equation becomes:

• \( \frac{d}{dx}(e^{2 \sin x}y) = \cos x e^{2 \sin x} \).

Recognizing this setup means that the left side can be directly integrated, which is a huge step in solving the differential equation. This process conveniently bundles the functions into one derivative, allowing for straightforward integration. Understanding this concept is key to simplifying seemingly complex differential equations and finding solutions efficiently.

When solving differential equations through integration, introducing a constant of integration, \( C \), is essential. This constant accounts for any constants that might have been present in the original function before differentiation. During integration, you will often see expressions like:

• \( e^{2 \sin x}y = \int \cos x e^{2 \sin x} \, dx + C \).

The constant of integration ensures that all possible function solutions are considered because the antiderivative could differ by a constant. When dividing to solve for \( y \), this \( C \) remains crucial as it provides the general solution. It's important to remember this constant can later be specified if initial conditions are provided. This ensures a unique solution that fits the given problem context.