Find Degree and Order of `((d^(2)y)/(dx^(3)))^(2) + cos ((dy)/(dx)) =0`

Answer

To find the degree and order of the given differential equation

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

we will follow these steps:

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

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

**Order = 2**

### Step 3: Identify 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, we have \(\left(\frac{d^2y}{dx^2}\right)^2\), which is raised to the power of 2. However, the term \(\cos\left(\frac{dy}{dx}\right)\) involves a trigonometric function of the first derivative, which makes the equation non-polynomial in terms of derivatives.

Since the equation involves the cosine function, the degree is not defined.

**Degree = Not Defined**

### Final Answer:
- **Order = 2**
- **Degree = Not Defined**

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