Question

The order and degree of the differential equation $$\frac{d^2y}{dx^2} = \sqrt{\frac{dy}{dx}}$$ are respectively

• A. 2, 3
• B. 3, 3
• C. 2, 2
• D. 1, 3

Hint:

Never look at the degree until you've completely cleared out fractional exponents and roots surrounding your derivative terms! Squaring or cubing early prevents falling into standard trap options.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to find the order and degree of the given differential equation containing a radical square root sign on the right-hand side.

Step 2: Key Formula or Approach:
The

order of a differential equation is the highest derivative present in the equation.
The

degree of a differential equation is the power of the highest order derivative, given that the differential equation is expressed as a polynomial in derivatives (free from fractional exponents and radicals).

Step 3: Detailed Explanation:
The given differential equation is:
$$\frac{d^2y}{dx^2} = \sqrt{\frac{dy}{dx}}$$ To determine the degree, we must first clear the radical (fractional exponent of $1/2$) by squaring both sides of the equation:
$$\left(\frac{d^2y}{dx^2}\right)^2 = \frac{dy}{dx}$$ Now, let's analyze the transformed polynomial equation:
1. The highest order derivative present is $\frac{d^2y}{dx^2}$, which means the $\text{order} = 2$.
2. The exponent power raised on this highest derivative term is 2, which means the $\text{degree} = 2$.
Therefore, the order and degree are 2 and 2 respectively, matching option (C).

Step 4: Final Answer:
The order and degree are respectively 2, 2, which corresponds to option (C).