find order and degree `(d^(2)y)/(dx^(2)) + (dy)/(dx) + x= sqrt(1+ (d^(3)y)/(dx^(3))`

Answer

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

\[ \frac{d^2y}{dx^2} + \frac{dy}{dx} + x = \sqrt{1 + \frac{d^3y}{dx^3}} \]

we can follow these steps:

### Step 1: Square both sides of the equation

To eliminate the square root, we square both sides of the equation:

\[ \left(\frac{d^2y}{dx^2} + \frac{dy}{dx} + x\right)^2 = 1 + \frac{d^3y}{dx^3} \]

### Step 2: Identify the highest order derivative

Now, we need to identify the highest order derivative present in the equation.

From the left-hand side, the highest derivative is \(\frac{d^2y}{dx^2}\), and from the right-hand side, the highest derivative is \(\frac{d^3y}{dx^3}\).

Thus, the highest order derivative in the entire equation is:

\[ \frac{d^3y}{dx^3} \]

### Step 3: Determine the order of the differential equation

The order of a differential equation is defined as the highest order of derivative present. Since the highest order derivative is \(\frac{d^3y}{dx^3}\), the order of the differential equation is:

\[ \text{Order} = 3 \]

### Step 4: Determine 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 a polynomial in derivatives.

In our case, the highest order derivative \(\frac{d^3y}{dx^3}\) appears to the power of 1 (it is not raised to any power other than 1). Therefore, the degree of the differential equation is:

\[ \text{Degree} = 1 \]

### Final Answer

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

- **Order:** 3
- **Degree:** 1

---