Find the order and degree, if defined, of the following differential equation :
`xy (d^(2)y)/(dx^(2))+x((dy)/(dx))^(2)-y(dy)/(dx)=0`

Answer

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

\[ xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0 \]

we will follow these steps:

### Step 1: Identify the highest order derivative
The first step is to identify the highest order derivative present in the equation.

In the given equation, we have:
- \( \frac{d^2y}{dx^2} \) (the second derivative)
- \( \frac{dy}{dx} \) (the first derivative)

The highest order derivative is \( \frac{d^2y}{dx^2} \).

### Step 2: Determine the order
The order of a differential equation is defined as the highest order of the derivative present in the equation.

Since the highest order derivative is \( \frac{d^2y}{dx^2} \), the order of the differential equation is:

**Order = 2**

### Step 3: Determine the degree
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 equation, the term involving the highest order derivative \( \frac{d^2y}{dx^2} \) appears as:

\[ xy \frac{d^2y}{dx^2} \]

Here, \( \frac{d^2y}{dx^2} \) is raised to the power of 1. Therefore, the degree of the differential equation is:

**Degree = 1**

### Final Answer
- **Order = 2**
- **Degree = 1**