Problem 105 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The differential equation representing the family of curves \(y^{2}=2 c(x+\sqrt{c})\) where \(c>0\), is a parameter, is of order and degree as follows: \(\quad\) (A) order 1 , degree 2 (B) order 1 , degree 1 (C) order 1 , degree 3 (D) order 2, degree 2

Step-by-Step Solution

Answer

Order 1, Degree 1 (Corresponds to Option B)

Step 1: Understanding the Problem

We are given the family of curves in the form \( y^2 = 2c(x + \sqrt{c}) \), where \( c \) is a parameter. We need to find the differential equation that represents this family and determine its order and degree.

Step 2: Find the Parameter Expression

Let's express \( c \) in terms of \( x \) and \( y \). From the equation \( y^2 = 2c(x + \sqrt{c}) \), we need to eliminate \( c \) from this equation. This expression will help us write the differential equation.

Step 3: Derivative and Elimination of Parameter

Differentiate the equation with respect to \( x \), treating \( y \) as a function of \( x \). The derived or implicit differentiation can help in eliminating the parameter \( c \).

Step 4: Formulating the Differential Equation

Use the derivative obtained to eliminate the parameter \( c \) from the original equation or its derivative, to form a differential equation.

Step 5: Determine Order and Degree

Determine the order and degree of the resulting differential equation. The order is determined by the highest derivative present, and the degree is the highest power of this derivative.

Key Concepts

Order and Degree in Differential Equations

When we talk about the order of a differential equation, we refer to the highest derivative present in the equation. The degree, on the other hand, is the highest power to which the highest derivative is raised, but this only applies once the differential equation is free from fractions and radicals involving derivatives.

Understanding the order and degree is essential because it helps in determining the complexity and the method used for solving the differential equation. Let's say a given differential equation has an order of 1. This means the equation involves a first derivative like \( \frac{dy}{dx} \) or \( y' \). If the degree is 2, this means the highest power of that derivative in the equation is 2. Here are some points to remember:

• Order: The highest derivative in the equation.
• Degree: The power of the highest derivative after removing fractions and radicals.

For the equation given in the original problem, determining the order and degree requires eliminating parameters and derivatives, leading us to better understand the structure of solutions.

Elimination of Parameters

Elimination of parameters in a family of curves is a key step in forming a differential equation from those curves. A parameter is an extra variable that adds flexibility to the equation, but removing it helps us obtain a formula involving only the main variables and their derivatives.

In the context of the given problem, we need to eliminate the parameter \( c \) from the expression \( y^2 = 2c(x + \sqrt{c}) \). By eliminating \( c \), we essentially move towards a differential form where our focus is on \( x \), \( y \) and their derivatives. The method of implicit differentiation often aids in this process:

• Express \( c \) in terms of other variables and their derivatives.
• Differentiate implicitly to relate \( x \), \( y \), and \( \frac{dy}{dx} \).
• Substitute back to remove \( c \) entirely.

By doing this, the differential equation becomes independent of \( c \) and realistically represents the family of curves in terms of \( x \) and \( y \) alone.

Implicit Differentiation

Implicit differentiation is a powerful technique used to differentiate equations where variables cannot be easily separated. It is especially useful when dealing with equations like those found in families of curves that include parameters.

For the exercise, we apply implicit differentiation to \( y^2 = 2c(x + \sqrt{c}) \). We treat \( y \) as a function of \( x \), and \( c \) as a function that embodies the curve's family.


Here's how implicit differentiation can guide us:

• Treat \( y \) and \( c \) as functions of \( x \).
• Differentiate both sides of the equation with respect to \( x \).
• The result will include \( y' = \frac{dy}{dx} \), leaving an equation still containing \( c \).
• Use algebraic manipulation or substitution to eliminate \( c \).

Ultimately, implicit differentiation helps us derive a differential equation that reflects the dynamics and relations of \( x \) and \( y \) without the need for explicitly solving for \( y \) or \( c \). It's a valuable strategy in many scenarios involving parametric forms and curve families.