Question

The sum of the order and degree of the differential equation: \[ x \left( \frac{d^2 y}{dx^2} \right)^{1/2} = \left( 1 + \frac{dy}{dx} \right)^{4/3} \] is:

• A. \( 5 \)
• B. \( 8 \)
• C. \( 12 \)
• D. \( 10 \)

Hint:

The order of a differential equation is the highest derivative present, while the degree is the exponent of the highest order derivative after clearing radicals or fractions. If the equation contains fractional exponents, express it in polynomial form before determining the degree.

Answer 1

The Correct Option is A

We are given the differential equation: \[ x \left( \frac{d^2 y}{dx^2} \right)^{1/2} = \left( 1 + \frac{dy}{dx} \right)^{4/3} \] Step 1: Identifying the Order The order of a differential equation is the highest derivative present in the equation. From the given equation, \[ x \left( \frac{d^2 y}{dx^2} \right)^{1/2} = \left( 1 + \frac{dy}{dx} \right)^{4/3} \] - The highest derivative present is \( \frac{d^2y}{dx^2} \). \[ \text{Order} = 2 \] Step 2: Identifying the Degree The degree of a differential equation is the exponent of the highest derivative after removing radicals and fractional powers. In the given equation, the second-order derivative appears as \( \left(\frac{d^2y}{dx^2}\right)^{1/2} \). To remove the square root (which is a fractional power), square both sides: \[ x^2 \left( \frac{d^2y}{dx^2} \right) = \left( 1 + \frac{dy}{dx} \right)^{8/3} \] Since the highest derivative term \( \frac{d^2y}{dx^2} \) now appears with an exponent of 1, the degree is: \[ \text{Degree} = 1 \] Step 3: Summing Order and Degree \[ \text{Order} + \text{Degree} = 2 + 1 = 3 \] Step 4: Correct Answer: (1) \ 5