The degree of the differential equation
`((d^(2)y)/(dx^(2)))+((dy)/(dx))^(2)=x sin((d^(2)y)/(dx))`, is

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To determine the degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the form of the differential equation
The given equation involves the second derivative \(\frac{d^2y}{dx^2}\) and the first derivative \(\frac{dy}{dx}\). It also includes a sine function, which complicates the polynomial nature of the equation.

**Hint**: Check if the equation can be expressed as a polynomial in terms of the derivatives.

### Step 2: Check if the equation is polynomial in terms of derivatives
For the degree of a differential equation to be defined, it must be expressible as a polynomial in its derivatives. In this case, we see that \(\frac{d^2y}{dx^2}\) appears inside a sine function, which is not a polynomial expression.

**Hint**: Remember that a polynomial can only include terms that are sums of powers of the variables (in this case, derivatives) multiplied by constants.

### Step 3: Conclusion about the degree
Since the term \(\sin\left(\frac{d^2y}{dx^2}\right)\) is not a polynomial in \(\frac{d^2y}{dx^2}\), the differential equation does not meet the criteria for having a defined degree.

**Hint**: If any derivative is involved in a non-polynomial function (like sine, cosine, exponential, etc.), the degree is not defined.

### Final Answer
The degree of the given differential equation is **not defined**.