Question

The order and degree of the differential equation \[ \frac{dy}{dx} + \left( \frac{d^2y}{dx^2} + 2 \right)^{\frac{1}{2}} + \frac{d^3y}{dx^3} + 5 = 0 \] are respectively:

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

Hint:

To find the order of a differential equation, locate the highest order derivative. To find the degree, ensure that the equation is polynomial in derivatives and identify the highest exponent of the highest order derivative.

Answer 1

The Correct Option is C

Step 1: Identifying the Order of the Differential Equation 
The order of a differential equation is the highest order derivative present in the equation. \[ \frac{dy}{dx} + \left( \frac{d^2y}{dx^2} + 2 \right)^{\frac{1}{2}} + \frac{d^3y}{dx^3} + 5 = 0. \] Here, the highest order derivative is \( \frac{d^3y}{dx^3} \), which means the order of the differential equation is: \[ \mathbf{3}. \] 

Step 2: Identifying the Degree of the Differential Equation 
The degree of a differential equation is defined as the highest exponent of the highest order derivative after removing radicals and fractions. In this equation, the term \( \left( \frac{d^2y}{dx^2} + 2 \right)^{\frac{1}{2}} \) contains a fractional power. To determine the degree, we must first eliminate this radical by squaring both sides. After squaring, the highest exponent of \( \frac{d^3y}{dx^3} \) (the highest order derivative) is found to be: \[ \mathbf{2}. \]

 Step 3: Conclusion 
Thus, the correct answer is: \[ \mathbf{2, 2}. \]