91Ó°ÊÓ

When working with first-order linear differential equations, the integrating factor is a crucial tool that simplifies the solving process. The integrating factor is a function, denoted usually by \( \mu(x) \), which helps in transforming a non-exact differential equation into an exact one, making it easier to solve.

For the equation of the form \( y' + P(x)y = Q(x) \), the integrating factor is given by:

• \( \mu(x) = e^{\int P(x) \, dx} \)

In our example, \( P(x) = -\frac{3}{x} \), thus the integrating factor is computed as:

• \( \mu(x) = e^{-3\ln|x|} = x^{-3} \)

This factor, \( x^{-3} \), is then used to multiply the entire equation, helping convert the differential equation into a form that can be solved by exact differentiation. Knowing how to find and apply the integrating factor correctly is essential for solving such differential equations efficiently.

Exact differentiation is a method used to solve certain differential equations once an integrating factor has been applied. It turns the equation into the derivative of a product, making integration straightforward. When the left side of the linear differential equation becomes a single derivative after multiplication by the integrating factor, we say that the differential equation is now in 'exact' form.

Consider our differential equation after applying the integrating factor \( x^{-3} \):

• \( y' x^{-2} - 3y x^{-3} = x^2 \)

This can be rewritten as:

• \( \frac{d}{dx}(y x^{-3}) = x^2 \)

This transformation indicates that the left-hand side is the exact differentiation of the product \( y x^{-3} \). Thus, integrating both sides becomes natural and straightforward, leading to finding the general solution of the differential equation. Recognizing when you have reached this form is key to solving these problems.

The constant of integration, commonly denoted as \( C \), emerges when you integrate a function. It represents the family of all possible curves that are solutions to a differential equation. Integration is an indefinite process without limits, so the constant of integration accounts for this variability.

In our problem, after integrating the exact differentiation:

• \( \int \frac{d}{dx}(y x^{-3}) \, dx = \int x^2 \, dx \)
• This results in \( y x^{-3} = \frac{x^3}{3} + C \)

Here, \( C \) captures all the possible solutions that fit the differential equation before additional conditions (like initial or boundary conditions) are applied to determine specific values. The role of \( C \) is critical, as it represents the solution's flexibility, allowing for an infinite number of potential answers based on starting conditions.