Find the order and degree of the 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:

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

### Step 1: Identify the Order
The order of a differential equation is determined by the highest derivative present in the equation.

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

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

### Step 2: Identify the Degree
The degree of a differential equation is determined by the power of the highest order derivative, provided that the equation is a polynomial in derivatives.

- In our equation, the highest derivative is \(\frac{d^2y}{dx^2}\) and it appears with a power of 1 (i.e., it is not raised to any power).
- The right-hand side of the equation involves \(\left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^4\), which is a polynomial expression and does not affect the degree of the highest derivative.

Since the highest order derivative \(\frac{d^2y}{dx^2}\) is to the power of 1, the degree of the differential equation is **1**.

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

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