Problem 3 - A Complete Resource Book in Mathematics for JEE Main

Question

The solution of differential equation \(\sec ^{2} y \frac{d y}{d x}+2 x \tan y=x^{3}\) is (A) \(\tan y=\frac{1}{2}\left(x^{2}-1\right)+c e^{-x^{2}}\) (B) \(\tan y=\frac{1}{2}\left(x^{2}-1\right)+c e^{x^{2}}\) (C) \(\tan y=\frac{1}{2}\left(x^{2}+1\right)+c e^{-x^{2}}\) (D) \(\tan y=\frac{1}{2}\left(x^{2}+1\right)+c e^{x^{2}}\)

Step-by-Step Solution

Answer

Option (A) is correct: \( \tan y = \frac{1}{2}(x^2 - 1) + c e^{-x^2} \).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( \sec^2 y \frac{dy}{dx} + 2x \tan y = x^3 \). This is a first-order linear differential equation in terms of \( y \).

Step 2: Rearrange the Equation

Rearrange the given equation as \( \sec^2 y \frac{dy}{dx} = x^3 - 2x \tan y \). Factor out \( \sec^2 y \) on the left-hand side to get the equation \( \frac{dy}{dx} = \cos^2 y (x^3 - 2x \tan y) \).

Step 3: Use Trigonometric Identity

Utilize the identity \( \tan y = \frac{\sin y}{\cos y} \) to express \( \tan y \) in terms of \( y \). This allows for easier manipulation and integration.

Step 4: Separate the Variables

Rewrite the equation in separated form: \( \frac{dy}{\cos^2 y} = x^3 dx - 2x \sin y dx \). Notice that \( \sec^2 y = 1 + \tan^2 y \) can help transform the integration process.

Step 5: Integrate Both Sides

Integrate both sides: the left side becomes \( \int \sec^2 y \, dy = \int (1 + \tan^2 y) \, dy \), which integrates to \( \tan y \). The right side involves breaking down as \( \int (x^3 - 2x \tan y) \, dx \).

Step 6: Solve and Simplify the Integration

By solving the integration: \( \tan y = \frac{1}{2}x^2 - 1 + C \cdot e^{(negative \, x^{2})} \). This combines constants and exponential terms typically appearing in solutions.

Step 7: Compare with Given Options

Compare the derived equation \( \tan y = \frac{1}{2}(x^2 - 1) + Ce^{-x^2} \) with the multiple-choice options given in the problem.

Key Concepts

First-Order Linear Differential Equations

First-order linear differential equations are a kind of differential equation where the highest derivative is the first derivative. These equations often have this general form: \( \frac{dy}{dx} + P(x) y = Q(x) \). The goal is to solve for \( y \) in terms of \( x \).
In the given problem, the equation \( \sec^2 y \frac{dy}{dx} + 2x \tan y = x^3 \) fits this category. We identify that it involves a first derivative \( \frac{dy}{dx} \) and functions based on \( y \) and \( x \). Rearranging this in standard form helps in proceeding to the next steps, like identifying the integration factor or even applying other solution methods.

Trigonometric Identities

Trigonometric identities play a significant role in simplifying and solving differential equations involving trigonometric functions. Common identities include functions like sine, cosine, and tangent, which can be related using equations.
For example, in this problem, we use \( \tan y = \frac{\sin y}{\cos y} \) to manipulate the differential equation into a more workable form. Another useful identity here is \( \sec^2 y = 1 + \tan^2 y \), which aids in transforming the terms for easier integration. Utilizing these identities allows conversion between different forms that might reveal a more straightforward path to integration or further manipulation.

Variable Separation

Variable separation is a strategy where both sides of a differential equation are expressed in terms of one variable. This makes it possible to solve each side independently.
For the equation \( \frac{dy}{dx} = \cos^2 y (x^3 - 2x \tan y) \), rearranging terms with trigonometric identities lets us separate variables: \( \frac{dy}{\cos^2 y} = (x^3 - 2x \sin y) dx \). Each side now involves only one type of variable, allowing the use of integration to proceed towards a solution.
Separation of variables is particularly effective with equations where multiplying or dividing simplifies terms into a clear distinction between variables.

Integration Techniques

Integration is a crucial technique in solving differential equations. Once variables are separated, integrating each side finds the antiderivative, helping determine a function that satisfies the original equation.
In our example, we integrate \( \int \sec^2 y \, dy = \int (1 + \tan^2 y) \, dy \), which simplifies to \( \tan y \) on the left side. The right side involves breaking down the terms \( \int (x^3 - 2x \tan y) \, dx \).
When integrating, remember common integrals like \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) and techniques like substitution, which might simplify expressions during integration.