Question

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

• A. \(2,\,2\)
• B. \(2,\,3\)
• C. \(3,\,4\)
• D. \(1,\,5\)

Hint:

To find the degree of a differential equation:

First remove radicals or fractions involving derivatives
Express the equation as a polynomial in derivatives
Degree is the power of the highest order derivative

Answer 1

The Correct Option is A

Step 1: Identify the order of the differential equation. The highest order derivative present in the equation is: \[ \frac{d^2y}{dx^2} \] Hence, \[ \text{Order} = 2 \] Step 2: Remove radicals to determine the degree. Given equation: \[ \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}} = k\left(\frac{d^2y}{dx^2}\right) \] To remove the fractional power, square both sides: \[ \left[1+\left(\frac{dy}{dx}\right)^2\right]^3 = k^2\left(\frac{d^2y}{dx^2}\right)^2 \] Step 3: Determine the degree. After removing radicals:

The equation is polynomial in derivatives
The highest order derivative is \(\dfrac{d^2y}{dx^2}\)
Its highest power is \(2\)
Therefore, \[ \text{Degree} = 2 \] Step 4: Final conclusion. The order and degree of the given differential equation are: \[ \boxed{2 \text{ and } 2} \]