91Ó°ÊÓ

An integrating factor is a function we use to simplify the solution of a linear first-order differential equation. By multiplying the entire differential equation by this factor, we can turn the left-hand side of the equation into an exact derivative of a product of functions, making it much easier to integrate.

To find the integrating factor, use the formula:
\[ \mu(x) = e^{\textstyle{\int P(x) \ dx}} \]
\( P(x) \) is derived from the standard form of the linear differential equation. In our example, we had:
\[ P(x) = -\frac{1}{x} \]
which leads to:
\[ \mu(x) = e^{\textstyle{\int -\frac{1}{x} \ dx}} = e^{\textstyle{-\ln|x|}} = \frac{1}{x} \]
This integrating factor transforms the equation into a form where it can be easily integrated and solved.

Differential equations come in different types, suitable for various methods of solution. Some common ones include:

• Separable Differential Equations: Can be written as \( f(y)dy = g(x)dx \).
• Linear First-Order: Of the form \( y' + P(x)y = Q(x) \).
• Linear Second-Order: Typically looks like \( y'' + P(x)y' + Q(x)y = R(x) \).

In the given problem, we identified the equation:
\[ x^{2} y^{\text{'}} - x y = \frac{1}{x} \]
as a linear first-order differential equation. It's recognized by the form \( y' + P(x)y = Q(x) \), characterized by its first-order derivative and linearity in both \( y \) and \( y' \), making it amenable to methods like using integrating factors.

Solving differential equations involves several steps tailored to the type of equation. For a linear first-order differential equation like the one in the exercise, the steps are:

• Identify the Equation Type: Determine if the differential equation is linear first-order, separable, etc.
• Rewrite in Standard Form: Express it in the form \( y' + P(x)y = Q(x) \).
• Find the Integrating Factor: Compute\( \mu(x) = e^{\textstyle{\int P(x) \ dx}} \).
• Multiply Through by the Integrating Factor: This simplifies it to the form \( d/dx (\mu(x)y) = \mu(x)Q(x) \).
• Integrate Both Sides: Find the solution by integrating both sides with respect to \( x \).
• Solve for \( y \): Solve the integral to find \( y \) explicitly.

In the exercise:
\[ \int (\frac{y}{x})^{\text{'}} dx = \int \frac{1}{x^{4}} dx \]
This solves to:
\[ y = x (\frac{x^{-3}}{-3} + C) = -\frac{1}{3x^{2}} + Cx \]
Each step builds on the previous one, leading to the solution of the differential equation.