Find the order and degree, if defined, of the following differential equation :
`(d^(2)y)/(dx^(2))=(2y^(3)+((dy)/(dx))^(4))/(sqrt((d^(2)y)/(dx^2)))`

Answer

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

\[ \frac{d^2y}{dx^2} = \frac{2y^3 + \left(\frac{dy}{dx}\right)^4}{\sqrt{\frac{d^2y}{dx^2}}} \]

we will follow these steps:

### Step 1: Identify the highest derivative
The highest derivative present in the equation is \(\frac{d^2y}{dx^2}\).

### Step 2: Determine the order
The order of a differential equation is defined as the highest order of derivative present. In this case, the highest derivative is \(\frac{d^2y}{dx^2}\), which is a second derivative. Therefore, the order of the differential equation is:

\[ \text{Order} = 2 \]

### Step 3: Rearrange the equation
To find the degree, we need to express the equation in a polynomial form. The given equation has a square root in the denominator, which complicates finding the degree directly. We can eliminate the square root by squaring both sides of the equation.

Squaring both sides gives:

\[ \left(\frac{d^2y}{dx^2}\right)^2 = \left(2y^3 + \left(\frac{dy}{dx}\right)^4\right) \]

### Step 4: Rearranging the equation
Now we can rearrange the equation:

\[ \left(\frac{d^2y}{dx^2}\right)^2 - \left(2y^3 + \left(\frac{dy}{dx}\right)^4\right) = 0 \]

### Step 5: Identify the highest power of the highest derivative
In the rearranged equation, the highest derivative is \(\frac{d^2y}{dx^2}\), and its highest power is 2 (from \(\left(\frac{d^2y}{dx^2}\right)^2\)).

### Step 6: Determine the degree
The degree of a differential equation is defined as the highest power of the highest order derivative when the equation is expressed as a polynomial in derivatives. Since the highest derivative \(\frac{d^2y}{dx^2}\) has a power of 2, the degree of the differential equation is:

\[ \text{Degree} = 2 \]

### Final Answer
Thus, the order and degree of the given differential equation are:

- **Order:** 2
- **Degree:** 2