Question

Determine the sum of the order and the degree of the following differential equation: \[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{frac{3}{2}} = k \frac{d^2y}{dx^2} \]

• A. \( 4 \)
• B. \( 3 \)
• C. \( 5 \)
• D. \( \text{The degree is undefined.} \)

Hint:

Never find the degree while fractional roots are still visible in the equation. Always clear out denominators in powers by raising the entire equation to the necessary scaling integer value first.

Answer 1

The Correct Option is A

Concept: The order of a differential equation is the order of the highest derivative present in the equation. The degree is the power of the highest-order derivative when the differential equation is written as a polynomial in its derivatives. This means all fractional exponents and radicals affecting the derivative terms must be cleared first.

Step 1: Clear the fractional radical exponent from the expression. The equation contains a fractional exponent of \( \frac{3}{2} \) on the left side. To turn this into a standard polynomial form, we square both sides of the entire equation: \[ \left(\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}\right)^2 = \left(k \frac{d^2y}{dx^2}\right)^2 \] \[ \left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = k^2 \left(\frac{d^2y}{dx^2}\right)^2 \]

Step 2: Identify the order and degree from the rationalized equation. 1. Look for the highest-order derivative present: The term \( \frac{d^2y}{dx^2} \) is a second-order derivative, so: \[ \text{Order} = 2 \] 2. Look for the exponent power attached to this highest-order derivative: The term \( \left(\frac{d^2y}{dx^2}\right) \) is raised to the power of 2, so: \[ \text{Degree} = 2 \]

Step 3: Calculate the sum of the two metrics. Adding our isolated values together: \[ \text{Sum} = \text{Order} + \text{Degree} = 2 + 2 = 4 \]