Problem 9 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The differential equation whose solution is \(A x^{2}+B y^{2}=1\) where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constants is of (a) second order and second degree (b) first order and second degree (c) first order and first degree (d) second order and first degree

Step-by-Step Solution

Answer

The differential equation is of first order and first degree.

Step 1: Differentiate the Given Solution

We start by differentiating the given solution equation \( A x^2 + B y^2 = 1 \) with respect to \( x \). This leads to:\[ 2Ax + 2By \frac{dy}{dx} = 0 \]

Step 2: Solve for \( \frac{dy}{dx} \)

From the differentiated equation \( 2Ax + 2By \frac{dy}{dx} = 0 \), isolate \( \frac{dy}{dx} \) by rearranging terms: \[ 2By \frac{dy}{dx} = -2Ax \]Divide both sides by \( 2By \): \[ \frac{dy}{dx} = - \frac{Ax}{By} \]

Step 3: Determine the Order and Degree

The expression \( \frac{dy}{dx} = - \frac{Ax}{By} \) is a first derivative, indicating that the differential equation is of first order. Additionally, since it doesn't involve any square roots or non-polynomial expressions of \( \frac{dy}{dx} \), it is of first degree as well.

Key Concepts

Order of Differential Equations

Understanding the order of differential equations is crucial for solving them. The order is determined by the highest derivative present in the equation. For instance, if a differential equation involves only the first derivative such as \( \frac{dy}{dx} \), then it is called a first-order differential equation. If it includes a second derivative like \( \frac{d^2y}{dx^2} \), it is a second-order equation.

When you differentiate the given solution \( A x^2 + B y^2 = 1 \), you obtain \( \frac{dy}{dx} = - \frac{Ax}{By} \), where \( \frac{dy}{dx} \) is the highest derivative. Thus, this specific equation is of first order.

Recognizing the order is essential as it tells you how many initial conditions you'll need for solving the equation. Each derivative represents a degree of freedom or an unknown variable that needs a boundary condition.

Degree of Differential Equations

The degree of a differential equation is another important concept. It is defined only for differential equations in which the derivatives appear in polynomial form. To find the degree, first make sure the equation is free from radicals and fractions as far as derivatives are concerned.

Once you have a polynomial equation, the degree is the power of the highest order derivative. For the equation \( \frac{dy}{dx} = - \frac{Ax}{By} \), the highest power of \( \frac{dy}{dx} \) is one, making it a first-degree differential equation.

Understanding this concept is helpful because it indicates how a solution might look and how complex it could be. Complicated degrees generally mean more intricate solutions, affecting methods for solving these equations.

Arbitrary Constants in Differential Equations

Arbitrary constants are symbols representing unspecified values in the solutions of differential equations. These constants (A and B in the exercise \( A x^2 + B y^2 = 1 \)), originate from the process of integration. Whenever you integrate a differential equation, you introduce a constant of integration.

The number of arbitrary constants in the solution of a differential equation equals the order of the equation. In solving differential equations, these constants can be determined if additional conditions, called boundary or initial conditions, are provided.

In this problem, the equation is of the first order, which aligns with having two arbitrary constants (A and B). These constants reflect the family's solutions, describing not just a single solution, but an entire set influenced by these parameters.