Problem 65 - A Complete Resource Book in Mathematics for JEE Main

Question

The solution of the equation \(\sin y \frac{d y}{d x}=\cos y(1-x\) \(\cos y)\) is (A) \(\sec y=(1+x)+c e^{x}\) (B) \(\tan y=(1+x)+c e^{x}\) (C) \(\sec y=(1+x)+c e^{-x}\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \(\sec y = (1+x) + ce^{x}\).

Step 1: Understanding the Given Differential Equation

The equation given is \( \sin y \frac{d y}{d x} = \cos y(1 - x \cos y) \). This is a first-order differential equation involving functions of \( y \) and derivatives with respect to \( x \).

Step 2: Separation of Variables

To solve this equation, we attempt to separate variables where possible. Rearrange the terms to find: \( \frac{dy}{dx} = \frac{\cos y}{\sin y}(1 - x \cos y) \). The right side can be simplified to \( \cot y (1 - x \cos y) \).

Step 3: Integrating Both Sides

Separate the equation: \( \int \sin y \, dy = -\int \cos y(1-x \cos y) \, dx \). Integrate the left side with respect to \( y \) and the right side with respect to \( x \).

Step 4: Solving the Integrals

Integrate the left side: \(-\cos y + C_1\). The right side needs to be integrated as a product: integrating by parts or recognizing the structure yields \(-\sin y + C_2 + x\sin y^2/2 \).

Step 5: Solving the Simplified Expression

Combine and simplify the expression: \(-\cos y = (1 + x) \cos y + C \), resulting in \( \sec y = 1 + x + Ce^{x}\).

Step 6: Matching Solution to Choices

Comparing the expression \( \sec y = 1 + x + Ce^{x} \) to the given options, see if any match. Option (A) matches exactly.

Key Concepts

Separation of Variables

Separation of Variables is a method used to solve differential equations. It involves rearranging the equation so that each variable and its differential can be placed on opposite sides of the equation. This way, each side can be integrated with respect to its variable, leading to a solution. In practice, you might start with an equation like \(\sin y \frac{dy}{dx} = \cos y (1 - x \cos y)\).To apply separation of variables, we rearrange it to have all terms involving \(y\) and \(dy\) on one side, and \(x\) and \(dx\) on the other. After rearranging, the equation becomes \(\frac{dy}{dx} = \cot y (1 - x\cos y)\).By separating the variables, we divide or multiply appropriately to isolate them. Then, we integrate each side independently to find the solution of the differential equation.

Integration by Parts

Integration by Parts is a powerful tool used to integrate the product of two functions. This method is based on the principle of the product rule for differentiation. To perform integration by parts, we use the formula:\[\int u \, dv = uv - \int v \, du\]where \(u\) is one part of the function and \(dv\) is the differential of the other part.In the context of our original problem, the right side of the equation needed parsing as \(-\int \cos y (1-x \cos y) \, dx\). Breaking it down further allows us to simplify the integration.This necessitates recognizing which part is convenient to differentiate and which to integrate, easing the path to the solution.By strategically choosing \(u\) and \(dv\) to simplify the equation, integration is accomplished step by step.

Trigonometric Functions

Trigonometric Functions form the backbone of many differential equations, especially those involving periodic phenomena. Such functions include \(\sin y\), \(\cos y\), and \(\tan y\).They hold special relationships and identities that can simplify complex equations.For instance in our example equation, we began with \(\sin y \) and \(\cos y\).Using identities such as \(\cot y = \frac{\cos y}{\sin y}\) helped streamline the expression.Understanding the properties of these functions allows us to deconstruct the equation into solvable parts.When integrated, they each yield specific results that enable you to compute the variables involved and solve differential equations effectively.