Order and degree of the differential equation ` (d^(2)y)/(dx^(2))+2(dy)/(dx) + sin y = 0 ` are

Options

A. order=1,degree =1
B. order=1,degree=not defined
C. order =2,degree=1
D. order =2,degree = not defined

Explanation

To determine the order and degree of the given differential equation:

\[ \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + \sin y = 0 \]

we will follow these steps:

### Step 1: Identify the highest derivative
The first step is to identify the highest derivative present in the differential equation.

In the equation, we have:
- \(\frac{d^2y}{dx^2}\) (the second derivative)
- \(\frac{dy}{dx}\) (the first derivative)

The highest derivative is \(\frac{d^2y}{dx^2}\).

### Step 2: Determine the order
The order of a differential equation is defined as the order of the highest derivative present in the equation.

Since the highest derivative is \(\frac{d^2y}{dx^2}\), which is a second derivative, the order of the differential equation is:

\[ \text{Order} = 2 \]

### Step 3: Determine the degree
The degree of a differential equation is defined as the power of the highest derivative when the equation is a polynomial in derivatives.

In our case, the highest derivative \(\frac{d^2y}{dx^2}\) appears to the power of 1 (it is not raised to any power other than 1). Therefore, the degree of the differential equation is:

\[ \text{Degree} = 1 \]

### Conclusion
Thus, the order and degree of the given differential equation are:

- **Order**: 2
- **Degree**: 1

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