Question

The degree of the differential equation \[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^{\frac{3}{4}} = \left( \frac{d^2y}{dx^2} \right)^{\frac{1}{3}} \]

• A. 4
• B. 9
• C. 6
• D. 2

Hint:

When finding the degree of a differential equation, ensure that all fractional exponents are eliminated by raising both sides of the equation to appropriate powers. The degree is then the highest power of the highest order derivative.

Answer 1

The Correct Option is A

Step 1: Identify the highest derivative.
The highest derivative in the given equation is the second derivative \( \frac{d^2y}{dx^2} \), which appears on the right-hand side of the equation.

Step 2: Remove fractional powers to find the degree.
To find the degree, we first need to remove any fractional powers from the equation. Start by eliminating the powers of \( \frac{3}{4} \) and \( \frac{1}{3} \).

Step 3: Raise both sides to appropriate powers to eliminate fractions.
We can raise both sides to the power of 4 to remove the fractional exponent on the left-hand side:
\[ \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^3 = \left( \frac{d^2y}{dx^2} \right)^4 \]

Step 4: Expand and simplify.
Now, the equation is simpler, and the highest power of the derivatives on the left-hand side is \( 3 \), and on the right-hand side, we have \( 4 \). Therefore, the degree of the equation is 4.

Step 5: Degree of the equation.
The degree of a differential equation is the highest power of the highest order derivative after removing fractional exponents. In this case, the highest power of the second derivative \( \frac{d^2y}{dx^2} \) is 4, so the degree is 4.

Step 6: Conclusion.
Thus, the degree of the given differential equation is \( 4 \), and the correct answer is option (A).