91Ó°ÊÓ

Implicit differentiation is a technique used when dealing with equations where the dependent variable \(y\) is not isolated on one side of the equation. For example, consider the equation:
\( -\frac{1}{x} + \frac{2}{x^2} + \frac{1}{y} - \frac{1}{y^2} = C \).
To find the derivative \( \frac{dy}{dx} \), we differentiate each term with respect to \(x\).

Why Use Implicit Differentiation?

• It helps when \(y\) can't be easily isolated.
• Allows for differentiation of complex relationships between variables.

When differentiating terms involving \(y\), apply the chain rule. Let's look at two terms separately:

• \( \frac{d}{dx} \big(\frac{1}{y} \big) = -\frac{1}{y^2} \frac{dy}{dx} \)
• \( \frac{d}{dx} \big( -\frac{1}{y^2} \big) = 2 \frac{1}{y^3} \frac{dy}{dx} \)

Notice that both derivatives involve \( \frac{dy}{dx} \). This is a key feature of implicit differentiation.

The chain rule is essential when differentiating composite functions. It states:

• If \(u = g(x)\) and \(y = f(u)\), then \( \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \).

Using the Chain Rule in Implicit Differentiation

The chain rule is applied whenever we differentiate an expression that is a function of another function. In the problem, terms involving \(y\) require the chain rule.
For example, in the term \( \frac{1}{y} \), \(y\) itself is a function of \(x\):

• Find the derivative of \( \frac{1}{y} \) with respect to \(y\).
• Multiply by \( \frac{dy}{dx} \) to account for the fact \(y\) is a function of \(x\).

Putting it all together:
\( \frac{d}{dx} \big( \frac{1}{y} \big) = -\frac{1}{y^2} \frac{dy}{dx} \)

A differential equation is an equation that involves derivatives of a function. It describes how a function changes and can often model real-world systems.
Given the equation:
\( \frac{dy}{dx} = \frac{(x-4)y^3}{x^3(y-2)} \)
This is a first-order differential equation because it involves the first derivative \( \frac{dy}{dx} \). Our task is to solve this equation.

General Solutions vs. Singular Solutions

• A general solution involves arbitrary constants and represents a family of curves.
• A singular solution is a specific solution that is not obtainable from the general solution and often serves as a boundary. In this case, \(y=0\) is being evaluated to see if it meets these criteria.

To determine if \(y=0\) is a solution, we substitute it into the differential equation and check:

• If the equation holds, \(y=0\) is a solution.
• If it satisfies the differential equation but not the general solution's form, it may be a singular solution.