Problem 111 - A Complete Resource Book in Mathematics for JEE Main

Question

The differential equation which represents the family of curves \(y=c_{1} e^{c_{x} x}\), where \(c_{1}\) and \(c_{2}\) are arbitrary constants is (A) \(y^{\prime}=y^{2}\) (B) \(y^{\prime \prime}=y^{\prime} y\) (C) \(y y^{\prime}=y^{\prime}\) (D) \(y y^{\prime}=\left(y^{\prime}\right)^{2}\)

Step-by-Step Solution

Answer

The correct answer is (D): \( y y' = (y')^2 \).

Step 1: Understand the Problem

We need to find the differential equation that represents the given family of curves, which is in the form \( y = c_1 e^{c_2 x} \), where \( c_1 \) and \( c_2 \) are arbitrary constants.

Step 2: Differentiate the Given Curve Once

Differentiate the curve with respect to \( x \). Since \( y = c_1 e^{c_2 x} \), the first derivative is \( y' = c_1 c_2 e^{c_2 x} \).

Step 3: Relate First Derivative to Original Function

Notice that \( y = c_1 e^{c_2 x} \) can be rewritten as \( y' = c_2 y \) by substituting \( y = c_1 e^{c_2 x} \) into the first derivative.

Step 4: Analyze the Multiple Choice Options

Compare the derived relation with the options given.- (A) \( y' = y^2 \) does not fit because \( y' = c_2 y \).- (B) \( y'' = y' y \) is not related because it involves a second derivative.- (C) \( y y' = y' \) simplifies to \( c_2 y^2 = c_2 y \), which is only true if \( y = 1 \), a particular case.- (D) \( y y' = (y')^2 \) holds since substituting gives \( y'(c_2 y) = (c_2 y)^2 \).

Step 5: Check Consistency with Option (D)

Verify that option (D) is consistent with our derived relation. Substituting \( y' = c_2 y \) back into (D), we get:\[ y \, c_2 y = (c_2 y)^2 \]Simplifying, we see both sides are equal, confirming it is consistent.

Step 6: Conclusion

The correct differential equation that represents the given family of curves is option (D): \( y y' = (y')^2 \).

Key Concepts

Family of Curves

In the world of differential equations, a family of curves refers to a set of curves that can be described with an equation involving arbitrary constants. These curves share a common characteristic, often observable through their shape or slope pattern. For the problem at hand, the family of curves given is described by the equation \( y = c_1 e^{c_2 x} \). Here, \( c_1 \) and \( c_2 \) are arbitrary constants, which means you can change their values to get different curves.
This framework of curves is significant in many ways:

• Each curve in the family can be considered as a specific solution to a differential equation.
• The change in constants can create infinitely many distinct curves, all belonging to the same family.
• Understanding the family helps in predicting behaviors of dynamic systems described by such equations.

Overall, a family of curves is a foundational concept that provides insight into how various solutions of differential equations are interrelated.

First Derivative

The first derivative of a function gives us valuable information about how the function is changing, in terms of its slope or rate of change. In the exercise, differentiating the curve \( y = c_1 e^{c_2 x} \) with respect to \( x \), yields its first derivative: \( y' = c_1 c_2 e^{c_2 x} \).
This tells us:

• The slope of the tangent to any curve described by the given equation at any point.
• How the curve increments or decrements; whether it's rising or falling at different points.
• Allows comparison to the original curve to establish relationships, such as in step 3 of the solution where \( y' = c_2 y \) is derived.

In many applications, knowing the first derivative is crucial for understanding the underlying dynamics of the system you're studying. For example, in physics, it can inform us about velocity when dealing with motion.

Order of Differential Equation

In the context of differential equations, the term 'order' refers to the highest derivative present in the equation. It's an important classification feature that tells us about the complexity and nature of the solutions we might expect.
For the given family of curves, the relevant differential equation deduced is \( y y' = (y')^2 \), which is a first-order differential equation as it involves only the first derivative \( y' \).
Here's what knowing the order helps with:

• Determining the methods appropriate for finding a solution to the equations.
• Indicating the number of arbitrary constants that should appear in the general solution.
• Simplifying the steps needed for solving the equation.

First-order equations are often simpler and have wide-ranging applications in various fields, including biology, economics, and physics.

Arbitrary Constants

Arbitrary constants play a pivotal role in understanding differential equations as they are indicative of the general solution's flexibility. In the provided exercise, constants \( c_1 \) and \( c_2 \), serve an important purpose.
To unpack their meaning:

• They allow for a family of solutions, collectively forming a general solution.
• They cater to initial conditions or specific scenarios by allowing the solution to be adapted based on provided data points.
• A first-order differential equation generally features one arbitrary constant, indicative of family solutions.

In practical terms, these arbitrary constants are the keys to personalizing a solution to fit specific boundary or initial conditions, thereby transitioning from a general to a particular solution.