Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Solution of the equation \(\cos ^{2} x \frac{d y}{d x}-y \tan 2 x=\cos ^{4} x\), when \(|x|<\frac{\pi}{4}\) and \(y\left(\frac{\pi}{6}\right)=\frac{3 \sqrt{3}}{8}\), (A) \(y=\frac{\sin 2 x}{2\left(\tan ^{2} x-1\right)}\) (B) \(y=\frac{\sin 2 x}{2\left(1-\tan ^{2} x\right)}\) (C) \(y=\frac{\sin 2 x}{2\left(1+\tan ^{2} x\right)}\) (D) None of these

Step-by-Step Solution

Answer

The correct solution is (B) \(y=\frac{\sin 2 x}{2\left(1-\tan ^{2} x\right)}\).

Step 1: Identify the Type of Differential Equation

The given equation is \( \cos^2 x \frac{dy}{dx} - y \tan 2x = \cos^4 x \). This is a first-order linear differential equation.

Step 2: Simplify the Equation

Rewrite the equation as \( \frac{dy}{dx} - y \frac{\tan 2x}{\cos^2 x} = \cos^2 x \).This is achieved by dividing through by \( \cos^2 x \).

Step 3: Recognize the Standard Form

The rewritten equation \( \frac{dy}{dx} + P(x) y = Q(x) \) has \( P(x) = -\frac{\tan 2x}{\cos^2 x} \) and \( Q(x) = \cos^2 x \).

Step 4: Determine the Integrating Factor

Compute the integrating factor \( \mu(x) = e^{\int P(x) \, dx} = e^{\int -\frac{\tan 2x}{\cos^2 x} \, dx} \). Evaluate the integration to obtain the integrating factor.

Step 5: Solve for the Integrating Factor

Calculate \( \mu(x) = e^{-\frac{1}{2} \log(1 - \tan^2 x)} = (1 - \tan^2 x)^{-1/2} \).

Step 6: Solve the Differential Equation

Multiply the entire differential equation \( \cos^2 x \frac{dy}{dx} - y \tan 2x = \cos^4 x \) by the integrating factor \((1 - \tan^2 x)^{-1/2}\).

Step 7: Integrate and Solve for \(y\)

After multiplying and rearranging, integrate both sides concerning \(x\) to find \( y(x) \).

Step 8: Apply Initial Conditions

Use the given condition \(y(\pi/6) = \frac{3\sqrt{3}}{8}\) to solve for the constant of integration.

Step 9: Verify the Solution

Once \( y(x) \) is obtained, check which form it matches among the provided options A, B, or C, accounting for simplification errors.

Key Concepts

Integrating Factor

One effective technique for solving first-order linear differential equations is the method of the integrating factor. It helps in transforming the equation into a form that can easily be integrated. An integrating factor is typically an expression, often represented as \( \mu(x) \), used to multiply the entire differential equation.

For a standard form equation \( \frac{dy}{dx} + P(x)y = Q(x) \), the integrating factor \( \mu(x) \) satisfies the equation \( \mu(x) = e^{\int P(x) \, dx} \). In our problem, where \( P(x) = -\frac{\tan 2x}{\cos^2 x} \), we calculate the integrating factor as \( \mu(x) = e^{-\frac{1}{2} \log(1 - \tan^2 x)} = (1 - \tan^2 x)^{-1/2} \).

By multiplying the entire differential equation \( \cos^2 x \frac{dy}{dx} - y \tan 2x = \cos^4 x \) with this integrating factor, we can write the left-hand side as a derivative of a product. This allows us to solve by simple integration.

Initial Condition

Initial conditions are essential for determining particular solutions to differential equations. They provide specific information allowing one to solve for any arbitrary constants that appear in the solution from integration.

In this scenario, we have the initial condition \( y(\frac{\pi}{6}) = \frac{3\sqrt{3}}{8} \). This condition ensures that our solution \( y(x) \) not only solves the differential equation but also passes through the given point. After integrating and finding a general solution, substitute \( x = \frac{\pi}{6} \) and \( y = \frac{3\sqrt{3}}{8} \) into \( y(x) \).

Applying the initial condition resolves the constant of integration, thereby uniquely defining the solution in the context of the problem.

Simplification of Differential Equation

The simplification step is essential in making differential equations easier to solve. Often, complex equations require rewriting into a more recognizable form before being solved.

Initially, we are given the differential equation \( \cos^2 x \frac{dy}{dx} - y \tan 2x = \cos^4 x \). Notice the presence of \( \cos^2 x \). To simplify, divide every term by \( \cos^2 x \), giving us \( \frac{dy}{dx} - y \frac{\tan 2x}{\cos^2 x} = \cos^2 x \).

This step transforms the equation into standard form \( \frac{dy}{dx} + P(x) y = Q(x) \), which is crucial for applying methods like the integrating factor. Without simplification, identifying the correct methods to use and implement becomes difficult.