Question

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

• A. 2, 2
• B. 2, 3
• C. 2, 1
• D. None of these

Hint:

The order of a differential equation is determined by the highest derivative, and the degree is determined by the exponent of the highest order derivative after making the equation polynomial in derivatives.

Answer 1

The Correct Option is A

We are given the differential equation: \[ \rho = \frac{\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2}}{\frac{d^2y}{dx^2}}. \] Step 1: Order of the differential equation.
The order of a differential equation is the highest order of the derivative present in the equation. In this equation, the highest derivative is \( \frac{d^2y}{dx^2} \), which is the second derivative of \( y \) with respect to \( x \). Therefore, the order of the differential equation is 2.
Step 2: Degree of the differential equation. The degree of a differential equation is the exponent of the highest order derivative, after the equation has been made polynomial in derivatives.
In this case, the equation is: \[ \rho = \frac{\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{3/2}}{\frac{d^2y}{dx^2}}. \] To find the degree, we need to remove the fractional powers of the derivatives. The expression \( \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^{3/2} \) is non-polynomial in \( \frac{dy}{dx} \), but we focus on the degree of the highest derivative \( \frac{d^2y}{dx^2} \). This derivative appears to the first power, and there are no fractional or negative exponents involving the highest derivative after simplification. Hence, the degree of the differential equation is 2.

Step 3: Conclusion. Therefore, the order of the differential equation is 2 and the degree is 2, so the correct answer is (a) 2, 2.