Question

The order and degree of the differential equation \(\mathrm{y}^{2}=\left[\left\\{1+(\mathrm{dy} / \mathrm{d} \mathrm{x})^{2}\right\\}^{(3 / 2)} /\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)\right]\) are (respectively) (A) 2,1 (B) 2,2 (C) 2,3 (D) 2,6

Short answer

The order of the given differential equation is 2, as the highest derivative present is \(\frac{d^2y}{dx^2}\). The degree of the given equation is 1, as this second derivative is raised to the power of 1 (in the denominator of a fraction). Therefore, the order and degree of the differential equation are (respectively) 2 and 1, which corresponds to option (A).

Step-by-step solution

Identify the order

The order of a differential equation is the highest derivative present. We can see two derivatives here: \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). The second derivative \(\frac{d^2y}{dx^2}\) has a higher order than the first derivative \(\frac{dy}{dx}\). Therefore, the order of the given equation is 2.

Identify the degree

The degree of a differential equation is the highest power of the highest order derivative present. In our case, the highest power of the second derivative \(\frac{d^2y}{dx^2}\) is raised to the power of 1, as it is in the denominator of a fraction. Thus, the degree of the given equation is 1. The order and degree of the given differential equation are (respectively) 2 and 1, which corresponds to option (A).

Key concepts

Order of Differential Equation

Understanding the order of a differential equation is fundamental for students, especially when preparing for competitive exams like JEE Maths. The order refers to the highest derivative present in the equation. For instance, in a basic differential equation like \( \frac{dy}{dx} = 3x \), the first derivative \( \frac{dy}{dx} \) is the highest order present, so the order is 1. For more complex equations like the one in our original exercise, we identify two derivatives: \( \frac{dy}{dx} \) and \( \frac{d^2y}{dx^2} \). The presence of the second derivative \( \frac{d^2y}{dx^2} \) indicates a second order differential equation.

In JEE Maths problems, correctly identifying the order is a crucial step because it guides the method of solution. Higher order equations might require advanced solutions such as using Laplace transforms or variation of parameters. Knowing the order also helps in determining the number of initial conditions or boundary values required to find a unique solution.

Degree of Differential Equation

Closely related to the concept of order is the degree of a differential equation. The degree is defined as the highest power of the highest order derivative after the equation has been made rational and polynomial in derivatives. For the given equation from the original exercise, some students might find it perplexing to see the second derivative \( \frac{d^2y}{dx^2} \) in the denominator. However, since our equation is already in a polynomial form with respect to \( \frac{d^2y}{dx^2} \) and its highest power is 1, the degree is identified as 1.

In simpler terms, if we have an equation where the highest order derivative is squared, such as \( \left(\frac{d^2y}{dx^2}\right)^2 \) , the degree would be 2. The intricate part is when the equation must be manipulated to clear out any radicals or fractions before finding the degree; this step must be performed cautiously to avoid errors. Degree serves as an important factor in the classification of differential equations, which in turn affects the solution methods students may employ.

Solving Differential Equations

The term solving differential equations refers to finding a function or set of functions that satisfy the given differential equation. There are several methods for solving differential equations, such as separation of variables, integrating factor method, or the use of homogeneous and linear equations.

For JEE Maths aspirants, mastering these methods is crucial as the exam often includes problems that test the ability to apply such techniques effectively. It's vital to start with simpler equations to build up a strong understanding before tackling more complex problems. The first step in solving any differential equation is to identify its order and degree, as these will influence which method to use. Also, understanding initial or boundary conditions is key to finding specific solutions to differential equations.

The method of solving will depend on the type of differential equation: for instance, ordinary differential equations (ODEs) involve single-variable functions, while partial differential equations (PDEs) deal with multivariable functions. JEE Maths covers ODEs extensively and expects students to be proficient at solving them.

JEE Maths

The Joint Entrance Examination (JEE) Maths section is renowned for its challenging and diverse range of questions, especially on the topic of differential equations. Students preparing for JEE are expected to have in-depth knowledge and skills in solving various types of differential equations since questions on this topic can significantly impact their overall score.

To excel in JEE Maths, students must focus on conceptual clarity and regular practice. It’s not just about memorizing formulas but understanding how to apply them in different scenarios. Visualizing the problems and solutions can also aid in comprehending the concepts more effectively. Using graphical methods, drawing slope fields, or understanding the behavior of solutions in real-world contexts can provide additional insights.

Moreover, time management and strategic guessing are also key strategies for approaching JEE Maths problems. With a combination of consistent practice and strategic test-taking, students can enhance their ability to quickly identify and solve differential equations on the JEE, thereby increasing their chances of achieving a top score.