91Ó°ÊÓ

Differential equations describe the relationship between a function and its derivatives, revealing how varying quantities change relative to each other. A **first-order differential equation** involves only the first derivative of a function. In our problem, it takes the form:

• \( x y' = 2y + x^2 \)

The equation above is linear, meaning it can have the structure \( y' + P(x)y = Q(x) \). Recognizing this means we can use techniques like integrating factors to find solutions. In this context, the derivative \( y' \) refers to how the function \( y \) changes with respect to \( x \). These equations are critical in modeling real-world phenomena where change is a primary factor, such as in physics, biology, and engineering.

The **integrating factor method** is a strategy to solve linear first-order differential equations. It transforms a non-exact equation into an exact one, which simplifies solving it. The original equation, \( y' - \frac{2}{x}y = x \), is first identified by rearranging terms to match the standard form, \( y' + P(x)y = Q(x) \). Once we see that:

• \( P(x) = -\frac{2}{x} \)
• \( Q(x) = x \)

The integrating factor, \( \mu(x) \), is computed as:

• \( \mu(x) = e^{\int P(x) \, dx} = x^{-2} \)

By multiplying the entire equation by this \( \mu(x) \), the equation becomes manageable:

• \( (x^{-2}y)' = x^{-1} \)

The left-hand side simplifies into a derivative of a product, enabling integration on both sides of the equation, which ultimately leads us to the solution.

**Initial conditions** are vital for particular solutions of differential equations. They specify values for the function at a certain point, allowing for the determination of the integration constant, \( C \), in the solution. In our exercise, we are given the initial condition \( y(1) = 3 \). Using it involves:

• Substituting \( x=1 \) and \( y=3 \) into the equation: \( y = x^2(\ln|x| + C) \)
• The substitution leads directly to: \( 3 = (1^2)(\ln 1 + C) = 0 + C \)
• Determining that \( C = 3 \).

This specific information allows us to write the exact solution for the equation as \( y = x^2(\ln|x| + 3) \). Initial conditions are crucial as they provide the needed leverage to resolve the arbitrary constants introduced during integration.