Question

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

• A. \( 3 \) and \( 12 \)
• B. \( 3 \) and \( 2 \)
• C. \( 3 \) and \( 1 \)
• D. \( 3 \) and \( 6 \)

Hint:

When finding the order and degree of a differential equation, first identify the highest derivative for the order, then check for any powers or fractional derivatives to determine the degree.

Answer 1

The Correct Option is C

The given differential equation is: \[ \frac{d^3 y}{dx^3} - 2 \left( \frac{d^2 y}{dx^2} \right) + xy = 0 \] 1. The order of a differential equation is the highest derivative present in the equation. Here, the highest derivative is \( \frac{d^3 y}{dx^3} \), so the order is 3. 2. The degree of a differential equation is the power of the highest derivative after removing any radicals or fractions. In this equation, the highest derivative is \( \frac{d^3 y}{dx^3} \), and it is not raised to any power other than 1. Therefore, the degree is 1. Thus, the order is 3, and the degree is 1. Hence, the correct answer is option (3).

Answer 2

Step 1: Identify the differential equation
\[ \frac{d^3 y}{dx^3} + 2 \left( \frac{d^2 y}{dx^2} \right) + xy = 0 \]

Step 2: Define order
The order of a differential equation is the highest order derivative present.
Here, the highest derivative is \( \frac{d^3 y}{dx^3} \), so the order is 3.

Step 3: Define degree
The degree is the power of the highest order derivative after removing any fractional or radical powers.
In this equation, all derivatives appear to the power 1 and are not under any radical or fraction.
So, the degree is 1.

Final Answer: Order = \( \boxed{3} \), Degree = \( \boxed{1} \)