91Ó°ÊÓ

When dealing with first-order linear differential equations, the integrating factor simplifies the process of finding a solution. For an equation in the form \( y' + P(x)y = Q(x) \), the integrating factor \( \mu(x) \) is given by the formula:

• \( \mu(x) = e^{\int P(x) \, dx} \)

In the given exercise, \( P(x) = 1 \). Thus, the integrating factor is:

• \( \mu(x) = e^{x} \)

By multiplying every term of the differential equation by this factor, you transform the equation into a form where the left-hand side becomes a derivative of a product. This enables us to recognize it as:

• \( \frac{d}{dx}(ye^x) = 2e^x \)

This product rule recognition simplifies the equation, making it easier to solve for \( y \). It’s an efficient method to solve linear differential equations.

The general solution provides a formula that describes all possible solutions of a differential equation. In our exercise, after manipulating the equation using the integrating factor, we reach the form:

• \( \frac{d}{dx}(y e^x) = 2e^x \)

To find the general solution, we integrate both sides. Integrating the right side:

• \( \int 2e^x \, dx \)
• leads to \( 2e^x + C \)

where \( C \) is the constant of integration. The solution to the original equation is then:

• \( y e^x = 2e^x + C \)

By dividing through by \( e^x \), we isolate \( y \):

• \( y = 2 + Ce^{-x} \)

This general solution encapsulates all solutions due to the inclusion of the constant \( C \).

The constant of integration \( C \) appears when we integrate a function, indicating there are infinitely many possible solutions. This is because integration is essentially the reverse of differentiation, and different functions can have the same derivative. In the context of our differential equation, it represents a family of solutions.
After integrating the equation \( \int 2e^x \, dx \), we add \( C \) to express this infinite range:

• \( y e^x = 2e^x + C \)

When solving for \( y \), the constant \( C \) remains part of the solution:

• \( y = 2 + Ce^{-x} \)

Different values of \( C \) will determine different particular solutions within the family. This constant allows us to include any initial condition to find a specific solution.