91Ó°ÊÓ

A first-order linear differential equation is a type of equation that involves the first derivative of an unknown function, which we usually denote as \( y' \). In this case, the equation has the standard form:

• \( y' = P(x)y + Q(x) \)

Here, \( P(x) \) and \( Q(x) \) are functions of \( x \) alone, or constants. In our specific example, the differential equation is given as:

• \( y' = 1 + x + y + xy \)

When analyzed, this equation can be rearranged to fit the standard form:

• \( y' = (1 + x)y + (1 + x) \)

When you see this form, know it is telling you there's a linear relationship between the derivative of \( y \) and the variables involved. We use techniques like finding an integrating factor to solve these equations.

An integrating factor is a device we use to solve first-order linear differential equations. It simplifies the integration of such equations, allowing us to take an equation that's difficult to solve and transform it into an equation that's easier to handle.The most typical formula used for finding the integrating factor \( \mu(x) \) is:

• \( \mu(x) = e^{\int P(x) \, dx} \)

In our solution, \( P(x) \) was identified as \( 1 + x \), leading to the integral:

• \( \int (1 + x) \, dx = x + \frac{x^2}{2} \)

From this, we can find the integrating factor:

• \( \mu(x) = e^{x + \frac{x^2}{2}} \)

Multiplying through the original equation by this integrating factor effectively allows us to convert the equation into an exact differential equation.

An implicit solution of a differential equation is one where the dependent variable (usually \( y \)) is not isolated on one side of the equation. Instead, both \( y \) and \( x \) appear on both sides of the equation, making it implicit.In our step-by-step solution, after finding the integrating factor, we ended up with this form:

• \( e^{x + \frac{x^2}{2}} y = I + C \)

Here, \( I \) is an expression for the integral of \( e^{x + \frac{x^2}{2}} (1 + x) \), which can't be expressed nicely in simple terms. Consequently, it remains implicit.This method of expressing the solution works well since it circumvents the need for exact expressions when they are infeasible to obtain. Plus, this format allows for numerical methods to approximate solutions as needed.

In contrast to implicit solutions, an explicit solution is one where the dependent variable \( y \) is expressed solely in terms of the independent variable \( x \) and a constant \( C \).The explicit form is usually preferred when possible because it provides a clear, direct relationship between \( y \) and \( x \). For our problem, even though we started with a complex equation, the solution process led to:

• \( y = \frac{I(x) + C}{e^{x + \frac{x^2}{2}}} \)

This is very close to an explicit solution. However, because \( I(x) \) itself doesn't have a simple form, it's common to maintain the solution in an implicit format.In fields requiring practical application (like engineering or physics), this form allows employing computational methods to compute precise values for \( y \) based on \( x \) and \( C \). This highlights the flexibility between seeking explicit and settling for implicit solutions depending on the context.