Question

If \( m' \) and \( n' \) are the degree and order respectively of the differential equation \( 1 + \left( \frac{dy}{dx} \right)^3 = \frac{d^2 y{dx^2} \), then the value of \( (m + n) \) is: }

• A. \( 4 \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( 5 \)

Hint:

To find the degree and order of a differential equation, identify the highest derivative for order and the highest power of that derivative for degree after clearing fractions and radicals.

Answer 1

The Correct Option is B

To determine the degree and order of the given differential equation \( 1 + \left( \frac{dy}{dx} \right)^3 = \frac{d^2 y}{dx^2} \), :
- The order of a differential equation is the highest derivative present in the equation.
- The degree of a differential equation is the highest power of the highest-order derivative after the equation has been made free of fractions and radicals (if any) in terms of the derivatives.
Step 1: Identify the Order
- The given equation contains \( \frac{d^2 y}{dx^2} \), which is the second derivative of \( y \) with respect to \( x \).
- There are no higher derivatives present.
- Therefore, the order (\( n' \)) is 2.
Step 2: Identify the Degree
- The highest-order derivative in the equation is \( \frac{d^2 y}{dx^2} \).
- The equation is \( 1 + \left( \frac{dy}{dx} \right)^3 = \frac{d^2 y}{dx^2} \).
- To find the degree, we need to express the equation such that the highest derivative is isolated and check the power of that derivative.
- Rearrange the equation:
\[ \frac{d^2 y}{dx^2} = 1 + \left( \frac{dy}{dx} \right)^3 \] - Here, \( \frac{d^2 y}{dx^2} \) is raised to the power of 1, and there are no fractions or radicals involving \( \frac{d^2 y}{dx^2} \). - The presence of \( \left( \frac{dy}{dx} \right)^3 \) (first derivative) does not affect the degree, as the degree is determined by the highest power of the highest-order derivative.
- Thus, the degree (\( m' \)) is 1.
Step 3: Compute \( m + n \)
- \( m' = 1 \) (degree)
- \( n' = 2 \) (order)
- Therefore, \( m' + n' = 1 + 2 = 3 \).
Verification
- The equation \( 1 + \left( \frac{dy}{dx} \right)^3 = \frac{d^2 y}{dx^2} \) is already in a form where the highest derivative \( \frac{d^2 y}{dx^2} \) is of degree 1, and the order is 2. The problem likely intends \( m \) and \( n \) to represent degree and order, and the sum \( m + n \) is 3, matching option (B). \[ \boxed{3} \]