Step 1: Understanding the Concept:
The classification of differential equations is fundamentally based on two properties: Order and Degree.
The Order of a differential equation is defined as the order of the highest derivative present in the equation.
For example, if the equation contains \( y'' \) (second derivative) as its highest derivative, the order is 2.
The Degree of a differential equation is the power to which the highest-order derivative is raised, provided the equation is written as a polynomial in its derivatives.
A crucial rule is that the degree can only be determined if the differential equation is free from radicals (roots) and fractional exponents affecting any of the derivative terms.
If fractional powers are present, they must be eliminated through algebraic rationalization before the degree can be identified.
Step 2: Key Formula or Approach:
Identify the highest derivative present to find the Order.
Rationalize the equation to remove fractional exponents.
Identify the exponent of the highest-order derivative in the rationalized form to find the Degree.
Calculate the sum: \( \text{Order} + \text{Degree} \).
Step 3: Detailed Explanation:
We start with the provided differential equation:
\[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} = k \frac{d^2y}{dx^2} \]
First, we look for the highest order of differentiation in the entire equation.
We see a first derivative \( \frac{dy}{dx} \) and a second derivative \( \frac{d^2y}{dx^2} \).
Since the highest derivative present is the second derivative, we conclude that:
\[ \text{Order} = 2 \]
Next, we need to find the degree. We notice that the term on the left side is raised to a fractional power of \( \frac{3}{2} \).
To clear this fractional exponent, we must square both sides of the equation:
\[ \left( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2} \right)^2 = \left( k \frac{d^2y}{dx^2} \right)^2 \]
Using the power rule \( (a^m)^n = a^{m \cdot n} \), the left side simplifies as:
\[ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = k^2 \left( \frac{d^2y}{dx^2} \right)^2 \]
Now that the equation is in polynomial form relative to its derivatives, we can determine the degree.
The highest-order derivative is \( \frac{d^2y}{dx^2} \), and its exponent in this rationalized equation is 2.
Thus:
\[ \text{Degree} = 2 \]
The problem asks for the sum of the order and the degree:
\[ \text{Sum} = \text{Order} + \text{Degree} = 2 + 2 = 4 \]
Step 4: Final Answer:
The sum of the order and the degree of the given differential equation is 4.
This corresponds to Option (A).