Question

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

• A. \( 4 \)
• B. \( 6 \)
• C. \( 5 \)
• D. \( 8 \)

Hint:

The order of a differential equation is determined by the highest derivative, while the degree is determined by the exponent of the highest order derivative after making it polynomial.

Answer 1

The Correct Option is B

Step 1: Identify the order of the differential equation
The given equation is: \[ \frac{d^y}{dx^t} = c + \left( \frac{d^y}{dx^t} \right)^{\frac{3}{2}}. \] The order of a differential equation is the highest order derivative present in the equation. Here, the highest order derivative present is \( \frac{d^y}{dx^t} \), thus: \[ \text{Order} = t. \] Step 2: Determine the degree of the equation
The degree of a differential equation is the exponent of the highest order derivative after it has been made polynomial in derivatives. Here, the term \( \left( \frac{d^y}{dx^t} \right)^{\frac{3}{2}} \) has a fractional exponent. To make it a polynomial, we would need to raise everything to the power of \( \frac{2}{3} \), which results in: \[ \text{Degree} = 2. \] Step 3: Compute the Sum
\[ \text{Order} + \text{Degree} = 4 + 2 = 6. \] Step 4: Final Answer
Thus, the sum of the order and degree of the given differential equation is: \[ \boxed{6}. \]