The degree of `(dy)/(dx)+cosy=0` is not defined true or false ?

Answer

To determine whether the statement "The degree of \(\frac{dy}{dx} + \cos y = 0\) is not defined" is true or false, we need to analyze the given differential equation step by step.

### Step 1: Identify the differential equation
The given equation is:
\[ \frac{dy}{dx} + \cos y = 0 \]

### Step 2: Understand the concept of degree
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made polynomial in derivatives.

### Step 3: Identify the highest order derivative
In the equation \(\frac{dy}{dx} + \cos y = 0\), the highest order derivative present is \(\frac{dy}{dx}\), which is a first-order derivative.

### Step 4: Determine the power of the highest order derivative
The power of the first-order derivative \(\frac{dy}{dx}\) in this equation is 1.

### Step 5: Check if the degree is well-defined
The degree of a differential equation is well-defined only when the equation can be expressed as a polynomial in the derivatives. In this case, while we have a first-order derivative, the presence of \(\cos y\) (which is a function of the dependent variable \(y\)) complicates the situation.

### Step 6: Conclusion about the degree
Since the equation involves a trigonometric function of the dependent variable \(y\), the degree of the differential equation cannot be determined in a conventional sense. Therefore, the degree is not well-defined.

### Final Answer
Thus, the statement "The degree of \(\frac{dy}{dx} + \cos y = 0\) is not defined" is **True**.

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