Find the order the degree, if defined, of the following differential equation and state whether they are linear or non-linear :
`xy(dy)/(dx)=((1+y^(2))/(1+x^(2)))(1+x+x^(2))`.

Answer

To solve the given differential equation and determine its order, degree, and linearity, we will follow these steps:

### Step 1: Identify the Order of the Differential Equation
The given differential equation is:
\[ xy \frac{dy}{dx} = \frac{(1+y^2)}{(1+x^2)}(1+x+x^2) \]

To find the order, we need to identify the highest derivative present in the equation. The highest derivative here is \(\frac{dy}{dx}\), which is the first derivative of \(y\) with respect to \(x\).

**Order**: 1

### Step 2: 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 term is \(\frac{dy}{dx}\), and it appears to the power of 1.

**Degree**: 1 (since \(\left(\frac{dy}{dx}\right)^1\))

### Step 3: Check for Linearity
A differential equation is linear if it can be expressed in the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \(P(x)\) and \(Q(x)\) are functions of \(x\) only.

Rearranging our equation:
\[ \frac{dy}{dx} = \frac{(1+y^2)(1+x+x^2)}{xy(1+x^2)} \]

This equation contains the term \(y^2\), which makes it non-linear because it cannot be expressed solely in terms of \(y\) and its first derivative.

**Linearity**: Non-linear

### Summary of Results
- **Order**: 1
- **Degree**: 1
- **Linearity**: Non-linear