Problem 6 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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

Step-by-Step Solution

Answer

The differential equation is (c) \( y y^{\prime\prime} = (y')^2 \).

Step 1: Differentiate the given curve with respect to x

The given family of curves is \( y = c_1 e^{c_2 x} \). Differentiate both sides with respect to \( x \) to find \( y' \):\[ y' = \frac{d}{dx}(c_1 e^{c_2 x}) = c_1 c_2 e^{c_2 x} \]

Step 2: Identify another expression for y

Notice that the expression for \( y \) given is \( y = c_1 e^{c_2 x} \). We can already identify that \( y' = c_2 y \). Thus, we relate \( y' \) back to \( y \).

Step 3: Differentiate y' to find y''

Differentiate \( y' = c_1 c_2 e^{c_2 x} \) again with respect to \( x \) to find \( y'' \):\[ y'' = \frac{d}{dx}(c_1 c_2 e^{c_2 x}) = c_1 c_2^2 e^{c_2 x} \]

Step 4: Express y'' in terms of y and y'

Since \( y = c_1 e^{c_2 x} \) and \( y' = c_2 y \), we substitute these into the expression for \( y'' \):\[ y'' = c_1 c_2^2 e^{c_2 x} = c_2^2 y \]Now, we express \( y'' \) in terms of \( y \) and \( y' \):\[ y'' = c_2 y' \]

Step 5: Formulate the differential equation

Substitute the expressions \( y = c_1 e^{c_2 x} \), \( y' = c_2 y \), and \( y'' = c_2 y' \) into the options provided.From \( y'' = c_2 y' \), we can derive\[ y'' = \frac{y'^2}{y} \]This matches expression (c): \( y y^{\prime\prime} = (y')^2 \).

Step 6: Verify the selected equation

Check that the derived equation \( y y'' = (y')^2 \) holds by substituting back:\( y = c_1 e^{c_2 x} \), \( y' = c_2 y \), and \( y'' = c_2^2 y \).Therefore, \( y y^{\prime\prime} = y (c_2^2 y) = (c_2 y)^2 = (y')^2 \), confirms (c) is right.

Key Concepts

Family of Curves

Imagine a set of curves, where each curve is described by a similar rule but has different specific parameters. This is called a family of curves. Each curve in the family is related to a general expression and takes on a unique shape based on different values given to the constants in the expression.
Let's consider the given equation: - The family of curves is represented by the equation \( y = c_1 e^{c_2 x} \).
Here, you will notice that both \( c_1 \) and \( c_2 \) are constants.
By altering these constants while maintaining the same equation form, you can derive different curves within this family.
These curves share fundamental characteristics due to their shared functional form. - For example, all declining curves in this family will share the same exponential decay pattern, but each will have different starting points or growth rates due to different constant values.

Arbitrary Constants

When dealing with equations, especially those related to families of curves, there are often constants that appear in the solution. These constants are termed **arbitrary constants**. They are essential because they allow for a multitude of specific solutions to a general differential equation.

Consider the constants \( c_1 \) and \( c_2 \) from the equation \( y = c_1 e^{c_2 x} \):

• \( c_1 \) determines the initial value or starting point of the curve. It shifts the curve up or down.
• \( c_2 \) influences the rate at which the exponential function grows or shrinks. It modifies the steepness of the curve.

These constants are termed arbitrary because their values aren’t fixed by the equation itself. Instead, these are defined through initial conditions or boundary values specific to a particular problem.
This flexibility allows the solution to fit many initial conditions by adjusting these constants accordingly.

Exponential Functions

Exponential functions are prominently featured in calculus and differential equations due to their unique properties.
An exponential function takes the form \( y = c_1 e^{c_2 x} \) where \( e \) represents the base of the natural logarithm, roughly equal to 2.718.

Exponential functions are noteworthy because:

• They represent processes where the rate of change is proportional to the current value, leading to either exponential growth or decay.
• The exponential function is the inverse of the natural logarithm function, tying various math areas together.

When differentiating exponential functions, the result is proportional to the function itself. This is evident in the derivative \( y' = c_2 y \), reflecting the self-replicating nature of exponential growth.
In mathematical modeling, exponential functions describe patterns in finance, biology (such as population growth), and physics (for radioactive decay and other phenomena).
Understanding these properties is crucial in solving and interpreting differential equations involving exponential functions.