Determine the order and degree or each of the following. Also, state whether they are linear or non-linear:
`x(dy)/(dx)+(3)/((dy)/(dx))=y^(2)`

Answer

To solve the given differential equation and determine its order, degree, and whether it is linear or non-linear, we will follow these steps:

### Step 1: Write down the given differential equation
The given equation is:
\[ x \frac{dy}{dx} + \frac{3}{\frac{dy}{dx}} = y^2 \]

### Step 2: Simplify the equation
To simplify the equation, we can multiply through by \(\frac{dy}{dx}\) to eliminate the fraction:
\[ x \left(\frac{dy}{dx}\right)^2 + 3 = y^2 \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ x \left(\frac{dy}{dx}\right)^2 - y^2 \frac{dy}{dx} + 3 = 0 \]

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

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

### Step 6: Determine if the equation is linear or non-linear
A linear differential equation can be expressed in the form:
\[ a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x) \frac{dy}{dx} + a_0(x) y = g(x) \]
where \(a_i(x)\) and \(g(x)\) are functions of \(x\) only, and the dependent variable \(y\) and its derivatives appear to the first power only.

In our case, since we have \(\left(\frac{dy}{dx}\right)^2\) and \(y^2\), the equation is not linear. Therefore, we conclude that:
\[ \text{The equation is non-linear.} \]

### Final Summary
- **Order**: 1
- **Degree**: 2
- **Type**: Non-linear