91Ó°ÊÓ

A differential equation is an equation that involves an unknown function and its derivatives. In this context, the given differential equation is \( y' = \frac{1 - y^2}{2(1 + xy)} \). This equation describes how the derivative of \( y \), which is \( y' \), relates to \( y \) and \( x \). For this problem, we are particularly interested in verifying that a given implicit relation defines a solution to this differential equation. An implicit solution means that the relationship between \( x \), \( y \), and possibly their derivatives, is not necessarily solved for the dependent variable, \( y \) in terms of \( x \). Instead, it is implicit in the form \( xy^2 + 2y - x = c \). To show that this relation is indeed a solution, we need to check if it satisfies the given differential equation.

The product rule is a vital differentiation rule used in calculus. It helps us differentiate functions that are products of two smaller functions. The rule is expressed as \((uv)' = u'v + uv'\). Here, \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives. In the given problem, we apply the product rule when differentiating the term \( xy^2 \). If you consider \( x \) as \( u \) and \( y^2 \) as \( v \), then the product rule helps us differentiate to \( x'(y^2) + x(2yy') \), simplifying to \( y + 2xy' \) since \( x' = 1 \). Remembering to apply the product rule correctly is crucial in obtaining the right derivative, which is important when analyzing differential equations.

Implicit differentiation is a technique used to differentiate equations that are not solved explicitly for one variable in terms of another. Consider our earlier equation, \( xy^2 + 2y - x = c \), which isn’t given in the form \( y = f(x) \). Instead, both \( x \) and \( y \) are intertwined. We use implicit differentiation to take the derivative of both sides with respect to \( x \). By doing this, we uncover the derivative \( y' \) even though \( y \) is not isolated. In the example, we started by differentiating each part of \( xy^2 + 2y - x \) term by term. It's important to carefully apply differentiation rules, such as the product rule and chain rule when necessary, to correctly solve for \( y' \). Implicit differentiation is invaluable for solving differential equations where explicit solutions are complex or impossible to isolate.

An integration constant, often denoted by \( c \), arises in solutions to differential equations when integrating derivatives. It represents the unknown constant added to the antiderivative of a function when integrating. This constant reflects the family of solutions possible for a differential equation since many functions can share the same derivative. In our implicit solution \( xy^2 + 2y - x = c \), \( c \) signifies any value that satisfies the equation given initial conditions or boundary constraints. It highlights the flexibility of the solution set, especially in a differential equation, where solving leads to a general form rather than a single solution. Integrating correctly and adding the integration constant ensures that you account for all possible scenarios described by the differential equation.