Find the order and degree of the differential equation `(d^(2)y)/(dx^(2))-y((dy)/(dx))^(2)-6y=0`.

Answer

To find the order and degree of the given differential equation

\[ \frac{d^2y}{dx^2} - y\left(\frac{dy}{dx}\right)^2 - 6y = 0, \]

we will follow these steps:

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

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

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

### Step 3: Determine the degree
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made a polynomial in derivatives.

In our equation, the highest derivative \(\frac{d^2y}{dx^2}\) appears to the first power (it is not raised to any power other than 1).

Thus, the degree of the differential equation is **1**.

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

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