Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

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 differential equation's solution doesn't match the provided options.

Step 1: Rewrite the Differential Equation

Given the differential equation: \( \sin y \frac{dy}{dx} = \cos y (1-x \cos y) \).First, separate the variables to make it easier to solve:\[ \frac{\sin y}{\cos y} dy = (1 - x \cos y) dx. \]This can be rewritten as:\[ \tan y \, dy = (1 - x \cos y) \, dx. \]

Step 2: Simplify using Trigonometric Identities

Recognize that \( \tan y = \frac{\sin y}{\cos y} \) leading to integration using trigonometric identities:Separate the equation as:\[ \frac{1}{\cos y} \sin y \, dy = (1 - x \cos y) \, dx. \]Or, simply\[ d(\tan y) = (1-x \cos y) \, dx. \]

Step 3: Attempt Integration

Integrate both sides of the equation:For the left side:\[ \int d(\tan y) = \tan y + C_1. \]For the right side:\[ \int (1 - x \cos y) \, dx = \int (1) \, dx - \int (x \cos y) \, dx. \]

Step 4: Evaluate Right-side Integration

Further solving the right-side integration components:\[ \int 1 \, dx = x \] and since \( \cos y \) is treated as a constant in its integration with respect to \( x \),\[ \int x \cos y \, dx = \frac{x^2}{2} \cos y + C_2. \]

Step 5: Final Form of the Solution

Combining the integrated parts, the solution becomes:\[ \tan y = x - \frac{x^2}{2}\cos y + C. \]This form fits no given options exactly but represents the closest actionable steps to derive general solutions.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a useful technique when dealing with integrals that involve square roots or certain forms, especially those that can leverage trigonometric identities. This approach simplifies the integration process by transforming the integral into a form that is easier to solve. In the provided exercise, trigonometric substitution helps rewrite the differential equation using trigonometric identities.
Recognizing the initial differential equation, \( \sin y \frac{dy}{dx} = \cos y (1 - x \cos y) \), involves trigonometric functions. Applying trigonometric identities such as \( \tan y = \frac{\sin y}{\cos y} \) can clarify the integration path.

• By rewriting \( \tan y \) in the differential equation, we transform the problem into an integrable form.
• This simplification allows for straightforward integration of \( \tan y \, dy \).
• The use of trigonometric substitution is invaluable for handling expressions with trigonometric functions, which would otherwise be challenging to integrate directly.

Integration Techniques

Integration is a fundamental part of solving differential equations. The exercise introduces two distinct integrals. The first, for \( d(\tan y) \), and the second, more complex integral on the right side of the equation. Let's break these down.
The integral of \( d(\tan y) \) is straightforward, resulting simply in \( \tan y \), with a constant of integration, \( C_1 \). This is because we integrate a derivative, simplifying our task.

• For the left-hand side: \( \int d(\tan y) = \tan y + C_1 \).

The right-hand side of the equation involves the integration of \( 1 - x \cos y \) with respect to \( x \). This complexity requires the separation into fundamental parts.

• The integral \( \int 1 \, dx \) is basic, resulting in \( x \).
• The term \( x \cos y \) treats \( \cos y \) as constant during integration with respect to \( x \), leading to \( \frac{x^2}{2} \cos y + C_2 \).
• Upon solving, combine these to complete the integration: \( x - \frac{x^2}{2} \cos y + C \).

Variable Separation

Separating variables is a classic method for simplifying and solving first-order differential equations. In this process, we rearrange the equation such that one variable and its derivative are on one side of the equation, while the other variable and its derivative are on the opposite side.
For the problem at hand, the goal was to isolate \( dy \) and \( dx \) to enable integration of each respective part.

• Initial separation yielded: \( \tan y \, dy = (1-x \cos y) \, dx \).
• This demonstrates a clear separation between terms involving \( y \) and \( x \), enabling the direct application of integration techniques.
• By isolating variables, the problem narrows to evaluating two integrals, each dependent solely on its own variable and its corresponding differential.

Variable separation, once achieved, transforms the problem into two manageable integrals rather than a complex differential equation, simplifying the pathway to obtaining the solution.