find order and degree of given differential equation :`x +(dy)/(dx) = sqrt(1+ ((dy)/(dx))^(2))`

Answer

To find the order and degree of the given differential equation \( x + \frac{dy}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ x + \frac{dy}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]

### Step 2: Square both sides
To eliminate the square root, we will square both sides of the equation:
\[ \left(x + \frac{dy}{dx}\right)^2 = 1 + \left(\frac{dy}{dx}\right)^2 \]

### Step 3: Expand the left-hand side
Expanding the left-hand side using the formula \((A + B)^2 = A^2 + 2AB + B^2\):
\[ x^2 + 2x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 = 1 + \left(\frac{dy}{dx}\right)^2 \]

### Step 4: Simplify the equation
Now, we can simplify the equation by subtracting \(\left(\frac{dy}{dx}\right)^2\) from both sides:
\[ x^2 + 2x\frac{dy}{dx} = 1 \]

### Step 5: Identify the highest order derivative
In the simplified equation \(x^2 + 2x\frac{dy}{dx} = 1\), the highest order derivative present is \(\frac{dy}{dx}\), which is the first derivative.

### Step 6: Determine the order and degree
- **Order**: The order of the differential equation is the highest derivative present. Here, the highest derivative is \(\frac{dy}{dx}\), which is of order 1.
- **Degree**: The degree of a differential equation is the power of the highest order derivative when the equation is a polynomial in derivatives. The term \(\frac{dy}{dx}\) appears to the power of 1.

### Final Answer
Thus, the order of the given differential equation is 1 and the degree is also 1.