91Ó°ÊÓ

The integrating factor is a crucial tool used for solving first-order linear differential equations. When you come across an equation like the one given, \( y' + 2y = \sin x \), it is essential to reformulate it into the standard form: \( y' + P(x)y = Q(x) \).

• Here, \( P(x) = 2 \) and \( Q(x) = \sin x \).
• The integrating factor, \( \mu(x) \), is constructed by calculating \( e^{\int P(x) \, dx} \).
• In this exercise, \( P(x) = 2 \), hence the integral becomes \( \int 2 \, dx = 2x \).
• Thus, the integrating factor is \( e^{2x} \).

This integrating factor serves a specific purpose: it allows us to transform the original differential equation into one that is easier to solve. By multiplying the entire equation by \( e^{2x} \), the left side transforms to a product of a derivatives, facilitating the integration process. This important step greatly simplifies solving the differential equation by turning it into a simpler integration problem.

Integration by parts is a technique derived from the product rule of differentiation, useful when integrating products of functions. Suppose you are faced with the integral \( \int e^{2x} \sin x \, dx \). Here's how you can apply integration by parts:

• Choose \( u = \sin x \) and \( dv = e^{2x} \, dx \).
• This choice leads to \( du = \cos x \, dx \) and \( v = \frac{1}{2} e^{2x} \).

The integration by parts formula is \( \int u \, dv = uv - \int v \, du \). Therefore:\[ \int e^{2x} \sin x \, dx = \frac{1}{2} \sin x \cdot e^{2x} - \int \frac{1}{2} e^{2x} \cos x \, dx\]Multiple applications might be necessary if the simplification leaves another integral. In such scenarios, it's crucial to maintain a clear organization of the calculation process.Integration by parts can be a bit tricky but ultimately breaks complex integrals into more manageable pieces, or even offers solvable equations through repeated use.

When integrating, a constant of integration, denoted as \( C \), is always added. This constant represents the family of potential solutions that can differ by a constant since differentiation of a constant is zero. In the process of solving the differential equation originally mentioned, this concept arises naturally.

• After using the integrating factor, solving integrals is integral to finding expressible solutions.
• In the specific case of \( \int e^{2x} \sin x \, dx \), we integrate twice using integration by parts.
• The result of these integrations is \( \frac{1}{5}e^{2x}(2\sin x - \cos x) \) plus a constant of integration, \( C \).

When you divide through by the integrating factor, \( C \) remains essential. In the final expression \( y = \frac{1}{5}(2\sin x - \cos x) + Ce^{-2x} \), \( Ce^{-2x} \) is the term that represents the infinite family of curves that satisfy the differential equation. This highlights that without any initial conditions, the constant \( C \) allows solutions to vary, providing flexibility until further conditions are specified.