Question

Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).

Hint:

The order is determined by the "highest dash" (e.g., y'', y''') while the degree is the exponent on that highest-order term. Ensure the equation is free from radicals or fractional powers of derivatives before determining the degree.

Answer 1

Step 1: Understanding the Concept:
- Order of a differential equation is the order of the highest derivative present in the equation.
- Degree of a differential equation is the highest power of the highest-order derivative, provided the equation is a polynomial in its derivatives.
Step 2: Key Formula or Approach:
1. Identify all the derivatives in the equation.
2. Find the highest order among them. This is the 'order'.
3. Find the power to which this highest-order derivative is raised. This is the 'degree'.
Step 3: Detailed Explanation or Calculation:
The given differential equation is:
\[ \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \sin x = 0 \] 1. Identifying the order:
The derivatives in the equation are \( \frac{d^2y}{dx^2} \) (second derivative) and \( \frac{dy}{dx} \) (first derivative).
The highest order of the derivatives is 2. Therefore, the order of the differential equation is 2.
2. Identifying the degree:
The equation is a polynomial in terms of its derivatives. The highest-order derivative is \( \frac{d^2y}{dx^2} \), and its power is 1.
Therefore, the degree of the differential equation is 1.
Step 4: Final Answer:
The order of the differential equation is 2 and the degree is 1.