Find the order and degree, if defined, of the following differential equation :
`x^(2)(d^(2)y)/(dx^(2))=[1+((dy)/(dx))^(2)]^(4)`

Answer

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

\[ x^2 \frac{d^2y}{dx^2} = \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^4 \]

we will follow these steps:

### Step 1: Rewrite the equation in standard form
To analyze the equation, we can rearrange it to bring all terms to one side:

\[ x^2 \frac{d^2y}{dx^2} - \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^4 = 0 \]

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

In our equation, the derivatives present are:
- \( \frac{d^2y}{dx^2} \) (second derivative)
- \( \frac{dy}{dx} \) (first derivative)

The highest derivative here is \( \frac{d^2y}{dx^2} \), which is a second derivative.

Thus, the **order** of the differential equation is **2**.

### Step 3: 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 equation, the highest order derivative is \( \frac{d^2y}{dx^2} \). In the rearranged form, it appears as:

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

Here, the term \( \frac{d^2y}{dx^2} \) is raised to the power of 1 (since it is multiplied by \( x^2 \), which is not a derivative term).

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

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