Find the sum of the order and degree of the differential equation: `(dy)/(dx)=x""(dx)/(dy)`.

Answer

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

### Step 1: Rewrite the Equation
Start with the given equation:
\[ \frac{dy}{dx} = x \frac{dx}{dy} \]

### Step 2: Cross-Multiply
Cross-multiply to eliminate the fractions:
\[ dy \cdot dy = x \cdot dx \cdot dx \]
This simplifies to:
\[ dy^2 = x \cdot dx^2 \]

### Step 3: Rearrange the Equation
Rearranging gives us:
\[ \frac{dy^2}{dx^2} = x \]
This can be expressed as:
\[ \left(\frac{dy}{dx}\right)^2 = x \]

### Step 4: Identify the Order
The order of a differential equation is determined by the highest derivative present. In our case, the highest derivative is \( \frac{dy}{dx} \), which is a first-order derivative. Therefore, the order of the differential equation is:
\[ \text{Order} = 1 \]

### Step 5: Identify the Degree
The degree of a differential equation is the power of the highest order derivative when the equation is a polynomial in derivatives. Here, we have:
\[ \left(\frac{dy}{dx}\right)^2 = x \]
The highest order derivative \( \frac{dy}{dx} \) is raised to the power of 2. Therefore, the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Step 6: Calculate the Sum of Order and Degree
Now, we find the sum of the order and degree:
\[ \text{Sum} = \text{Order} + \text{Degree} = 1 + 2 = 3 \]

### Final Answer
Thus, the sum of the order and degree of the differential equation is:
\[ \boxed{3} \]