Problem 77 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The integrating factor of the differential equation \(\left(x^{2}-1\right) \frac{d y}{d x}+2 x y=x\) is (a) \(\frac{1}{x^{2}-1}\) (b) \(x^{2}-1\) (c) \(\frac{x^{2}-1}{x}\) (d) \(\frac{x}{x^{2}-1}\)

Step-by-Step Solution

Answer

The integrating factor is (b) \(x^2 - 1\).

Step 1: Write the Differential Equation in Standard Form

The given differential equation is \((x^{2}-1) \frac{dy}{dx} + 2xy = x\). Let's first express it in the standard linear form of a first-order differential equation: \(\frac{dy}{dx} + P(x)y = Q(x)\). Divide the entire equation by \(x^2 - 1\):\[\frac{dy}{dx} + \frac{2x}{x^2-1}y = \frac{x}{x^2-1}\]Here, \(P(x) = \frac{2x}{x^2-1}\) and \(Q(x) = \frac{x}{x^2-1}\).

Step 2: Determine the Integrating Factor

The integrating factor \(\mu(x)\) is found by the formula \(\mu(x) = e^{\int P(x)\,dx}\). Substitute \(P(x) = \frac{2x}{x^{2}-1}\) and find:\[\int P(x) \, dx = \int \frac{2x}{x^{2}-1} \, dx\]This integral is simplified by substitution. Let \(u = x^2 - 1\), then by substituting, \(du = 2x \, dx\). Therefore, the integral becomes:\[\int \frac{2x}{x^{2}-1} \, dx = \int \frac{1}{u} \, du = \ln|u| = \ln|x^{2}-1|\]Thus, the integrating factor is:\[\mu(x) = e^{\ln|x^2 - 1|} = |x^2 - 1|\]Since \(x^2 - 1\) is positive for most domains of \(x\), we take \(\mu(x) = x^2 - 1\).

Step 3: Choose the Correct Answer

From Step 2, we determined the integrating factor to be \(x^2 - 1\). Therefore, the correct answer is option (b): \(x^2 - 1\).

Key Concepts

Differential Equation

A differential equation is a mathematical expression involving an unknown function, its derivatives, and variables. These equations describe a wide variety of phenomena, such as motion, growth, and decay, in diverse fields like physics, engineering, and biology. Differential equations can be classified into several types based on their order, linearity, and homogeneity.

Key points to understand about differential equations include:

• The "order" refers to the highest derivative present in the equation. For instance, if the highest derivative is the second derivative, it’s a second-order differential equation.
• They can be either ordinary (ODE) or partial (PDE), with ODEs involving functions of a single variable and their derivatives, while PDEs involve multiple variables.
• Solutions to differential equations are functions that satisfy the equation. Finding them often involves integration and sometimes requires specific boundary or initial conditions.

Understanding how to manipulate and solve differential equations is crucial for modeling real-world situations effectively.

Linear Differential Equation

A linear differential equation is a specific type of differential equation where the unknown function and its derivatives appear linearly. This means that the function and its derivatives are not raised to any power other than one and are not multiplied together. Such equations are essential as they frequently model real-life systems that are proportional and additive.

The general form of a first-order linear differential equation is:\[\frac{dy}{dx} + P(x)y = Q(x)\]where:

• \(P(x)\) and \(Q(x)\) are functions of \(x\).
• \(y\) is the unknown function of \(x\).

To solve a linear differential equation, one typical method is finding an integrating factor (discussed in the next section), ensuring a systematic approach to find the solution. Linear equations are easier to tackle compared to non-linear ones because their solutions can be added to form more solutions, which is a property known as superposition.

Integrating Factor Formula

The integrating factor is a vital tool in solving linear differential equations. It is a strategic function that makes an equation easily integrable, allowing you to rewrite the equation in a form that is straightforward to solve. The integrating factor is particularly useful because it transforms the differential equation into a perfect derivative, simplifying the integration process.

The integrating factor \(\mu(x)\) is defined by the formula:\[\mu(x) = e^{\int P(x)\, dx}\]where:

• \(P(x)\) is the coefficient function of \(y\) in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\).
• \(\int P(x)\, dx\) is the integral of \(P(x)\).

By multiplying every term in the equation by \(\mu(x)\), the left-hand side becomes the derivative of a product, enabling you to integrate effectively. Solving the resulting equation yields the solution to the original differential equation. The integrating factor is a powerful formulaic approach that greatly simplifies solving linear differential equations.