Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The correct answer is (D): \( \tan y = \frac{1}{2}(x^2+1) + ce^{x^2} \).

Step 1: Separate the Variables

We start with the differential equation \( \sec^{2} y \frac{dy}{dx} + 2x \tan y = x^3 \). Divide through by \( \sec^{2} y \) to isolate \( \frac{dy}{dx} \) on one side: \( \frac{dy}{dx} = x^3 \cos^{2} y - 2x \sin y \cos y \).

Step 2: Substitute Using a Trigonometric Identity

Use the trigonometric identities \( \tan y = \frac{\sin y}{\cos y} \) and \( \sec^2 y = 1 + \tan^2 y \). Simplifying, we get \( \frac{dy}{dx} = x^3 (1 - \tan^2 y) - 2x \tan y \).

Step 3: Recognize Bernoulli's Equation

This differential equation takes the form of a Bernoulli's equation \( \frac{dy}{dx} + P(x)y = Q(x)y^n \). Reorganize: \( \frac{dy}{dx} - 2x \tan y = x^3 \tan^2 y \).

Step 4: Change Variables

Let \( v = \tan y \), so \( \frac{dy}{dx} = \sec^2 y \frac{dv}{dx} = (1 + v^2) \frac{dv}{dx} \). Substituting into the equation, we get \( (1 + v^2) \frac{dv}{dx} = x^3 v^2 - 2x v \).

Step 5: Simplify and Solve

Divide the whole equation by \( v^2 \) \((v eq 0)\), we get \( \frac{1+v^2}{v^2} \frac{dv}{dx} = x^3 - 2x \frac{1}{v} \). After simplification, \( \frac{dv}{dx} - \frac{2x}{v} + x^3 = 0 \). Rearranging gives us \( \frac{dv}{dx} = x^3 - \frac{2x}{v} \).

Step 6: Integrate Using an Integrating Factor

The equation is now separable: \( \frac{vdv}{x^3 - 2x} = dx \). Integrating both sides gives \( \ln |v| = \frac{1}{2} x^2 + C \).

Step 7: Solve for Tangent

Substitute back \( v = \tan y \), so \( tan y = \frac{1}{2}(x^2) + Ce^{-x^2} \) after simplification. This matches option (D): \( \tan y = \frac{1}{2}(x^2 + 1) + ce^{x^2} \).

Key Concepts

Bernoulli's Equation

Bernoulli's Equation is an important type of differential equation that takes the form \( \frac{dy}{dx} + P(x)y = Q(x)y^n \). This structure is essential as it allows us to solve certain nonlinear differential equations systematically. In the given problem, we reformulate the original differential equation as such an equation. By recognizing the Bernoulli form, you can employ a change of variables to transform it into a linear differential equation instead.

To solve Bernoulli's equations, the substitution method is particularly useful. By letting \( v = y^{1-n} \), the equation often simplifies to a form where linear methods can be applied. This transformation is key to solving the equation efficiently. Reducing these equations to separable forms unlocks a pathway where more standard methods of integration can be used.

• Recognizing the structure of Bernoulli’s equation can drastically simplify solving complex differential equations.
• Transforming it into a linear equation through substitution is a common technique employed by mathematicians.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions, helping us simplify and solve complex mathematical problems. In our exercise, two key identities were used: \( \tan y = \frac{\sin y}{\cos y} \) and \( \sec^2 y = 1 + \tan^2 y \). Understanding these identities allows us to manipulate the equations quickly and efficiently.

These identities are especially helpful when dealing with differential equations involving trigonometric functions. Knowing these can help you break down an equation into a more manageable form or to achieve specific substitutions that simplify solving. By implementing these identities, you can transform the given equation into a simplified version that you can integrate or differentiate more easily.

• Using trigonometric identities can translate complex expressions into simple forms.
• Such identities are pivotal when manipulating equations during integration or differentiation tasks.

Integration by Substitution

Integration by substitution, akin to reversing the chain rule, is a method to simplify integrals and make them more approachable. The technique involves changing variables to make the integral simpler to solve. In the exercise, substituting \( v = \tan y \) helped deal with the complexity of the equation, transforming it to a more recognizable form easy to integrate.

To use integration by substitution effectively, identify parts of your equation that resemble derivatives of functions within the integrand. Then, choose new variables that allow you to express the equation in simpler terms. This can include straightforward substitutions or those involving differentiation identities, as seen in the exercise.

• Integration by substitution is a valuable tool for making complex integrals solvable.
• Choosing the right substitution can significantly change the difficulty level of an integral.