The order and degree of the differential equation `(d^(2)y)/(dx^(2))+cos((dy)/(dx))=0` respectively are

Options

A. both degree and other can not be defined
B. order =2, degree can not be defined
C. 2 and 1
D. degree =2, order can not be defined

Answer

To determine the order and degree of the given differential equation

\[ \frac{d^2y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0, \]

we will follow these steps:

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

In this equation, we have:
- The term \(\frac{d^2y}{dx^2}\) which is the second derivative of \(y\) with respect to \(x\).
- The term \(\frac{dy}{dx}\) which is the first derivative of \(y\) with respect to \(x\).

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

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

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

\[ \text{Order} = 2. \]

### Step 3: Determine the degree
The degree of a differential equation is defined when the equation can be expressed in polynomial form in terms of the derivatives.

In our equation, we have:

\[ \cos\left(\frac{dy}{dx}\right), \]

which is a trigonometric function of the first derivative \(\frac{dy}{dx}\). Since this term is not a polynomial in \(\frac{dy}{dx}\), the degree of the differential equation is not defined.

### Conclusion
Thus, we conclude that:
- The order of the differential equation is \(2\).
- The degree of the differential equation is not defined.

### Final Answer
The order and degree of the differential equation are \(2\) and not defined, respectively.

---