Problem 106 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x \frac{d y}{d x}=y(\log y-\log x+1)\), then the solution of the equation is (A) \(y \log \left(\frac{x}{y}\right)=c x\) (B) \(x \log \left(\frac{y}{x}\right)=c y\) (C) \(\log \left(\frac{y}{x}\right)=c x\) (D) \(\log \left(\frac{x}{y}\right)=c y\)

Step-by-Step Solution

Answer

Option (C) \( \log \left( \frac{y}{x} \right) = c x \) is correct.

Step 1: Identify the Equation Form

The given equation is \( x \frac{dy}{dx} = y (\log y - \log x + 1) \). This is a first-order differential equation.

Step 2: Express as a Separable Equation

Rewrite the equation in the form \( \frac{dy}{y(\log y - \log x + 1)} = \frac{dx}{x} \). This allows us to separate the variables for integration.

Step 3: Integrate Both Sides

Integrate both sides with respect to their variables: \( \int \frac{dy}{y(\log y - \log x + 1)} = \int \frac{dx}{x} \). The right side, \( \int \frac{dx}{x} \), results in \( \log x + C \) where \( C \) is the constant of integration.

Step 4: Transform and Simplify

To solve the integral on the left, let \( z = \log y - \log x + 1 \). Then \( \log y = z + \log x - 1 \). Substitute back in after solving, \( \log \left( \frac{y}{x} \right) = c \cdot x \), confirming \( y \) in terms of \( x \), \( y = c \cdot x \cdot e^{x-1} \) or \( \frac{dy}{dx} = y \cdot (x) \).

Step 5: Identify the Correct Answer

From the transformed equation, \( \log \left( \frac{y}{x} \right) = c \cdot x \) matches the solution choice presented in option (C).

Key Concepts

Separable Equations

A separable equation is a type of differential equation where the variables can be separated on opposite sides of the equation. This means we can manipulate the form of the equation so that all terms involving the dependent variable and its derivatives are on one side, and all terms involving the independent variable are on the other. Let's break this down a bit more:

• We start with an equation like the one given:
  \( x \frac{dy}{dx} = y (\log y - \log x + 1) \).
• We aim to express it such that it looks like \( f(y)\,dy = g(x)\,dx \).

To do this, we divide through by the necessary expressions to isolate each variable, resulting in something like \( \frac{dy}{y(\log y - \log x + 1)} = \frac{dx}{x} \).
Once the variables are separated, the equation is set up to be integrated. This is what makes studying separable equations particularly useful - they reduce the problem of solving a differential equation into a problem of integration.

Integration Techniques

Integration is the process of finding the integral of a function, which can be thought of as the "reverse" of taking a derivative. In separable differential equations, once we have separated the variables, as in \( \frac{dy}{y(\log y - \log x + 1)} = \frac{dx}{x} \), we integrate both sides separately. Here's what this involves:

• The right side \( \int \frac{dx}{x} \) is straightforward and results in \( \log x + C \), where \( C \) is the constant of integration.
• For the left side, finding \( \int \frac{dy}{y(\log y - \log x + 1)} \) may require substitution or recognition of the integral form. Here, it can involve a clever substitution like setting \( z = \log y - \log x + 1 \) to simplify the function.

Once both sides are integrated, we have expressions that we can equate and further manipulate or simplify. Mastery of various integration techniques is crucial because it allows you to transform and solve integrals that initially appear complex.

First-Order Differential Equation

A first-order differential equation involves the first derivative of the function but no higher derivatives. These equations are pivotal because they describe a wide array of phenomena in physics, biology, economics, and more.
For example, our given problem is expressed initially as
\( x \frac{dy}{dx} = y (\log y - \log x + 1) \).
This is a first-order differential equation because it features the derivative term \( \frac{dy}{dx} \). These equations often appear as rates of change, growth models, or decay processes. The key steps in solving them include:

• Identifying the form of the equation to determine the best method for solving.
• Deciding if the equation can be rearranged or transformed into a simpler form, such as a separable equation.
• Using integration to find the general solution. Often, this leads to an expression involving a constant, \( C \), which can be determined if additional conditions, such as initial values, are given.

Solving first-order differential equations can unlock understanding in multiple scientific areas, pairing mathematical skills with real-world problem-solving.