The degree and order respectively of the differential equation `(dy)/(dx) = (1)/(x + y + 1)` are

Options

A. 1 , 1
B. 1 , 2
C. 2 , 1
D. 2 , 2

Answer

To determine the degree and order of the given differential equation \( \frac{dy}{dx} = \frac{1}{x + y + 1} \), we will follow these steps:

### Step 1: Identify the order of the differential equation
The order of a differential equation is defined as the highest order of the derivative present in the equation.

In our equation, we have:
\[ \frac{dy}{dx} \]
This is a first-order derivative. Therefore, the order of the differential equation is:
\[ \text{Order} = 1 \]

### Step 2: Identify the degree of the differential equation
The degree of a differential equation is defined as the power of the highest order derivative when the equation is expressed as a polynomial in terms of the derivatives.

In our case, the highest order derivative is \( \frac{dy}{dx} \), and it appears to the power of 1 (since it is not raised to any power, we consider it as 1).

Thus, the degree of the differential equation is:
\[ \text{Degree} = 1 \]

### Conclusion
Putting it all together, we find that:
- The order of the differential equation is 1.
- The degree of the differential equation is 1.

Therefore, the degree and order respectively of the differential equation \( \frac{dy}{dx} = \frac{1}{x + y + 1} \) are:
\[ \text{Order} = 1, \quad \text{Degree} = 1 \]

### Final Answer
The degree and order respectively of the differential equation are \( 1, 1 \).

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