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 find the degree of the 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 order of the differential equation
The highest derivative present in the equation is \(\frac{d^2y}{dx^2}\), which is a second-order derivative. Therefore, the order of the differential equation is 2.

**Hint:** The order of a differential equation is determined by the highest derivative present in the equation.

### Step 2: Analyze the terms in the equation
The left side of the equation consists of \(\frac{d^2y}{dx^2}\) and \(\left(\frac{dy}{dx}\right)^2\). The right side includes \(x \sin\left(\frac{d^2y}{dx^2}\right)\), where the sine function is applied to the second-order derivative.

**Hint:** Look for the highest power of the derivatives in the equation and how they are combined.

### Step 3: Determine if the equation is polynomial in derivatives
For the degree of a differential equation to be defined, the equation must be a polynomial in its derivatives. In this case, the term \(\sin\left(\frac{d^2y}{dx^2}\right)\) is not a polynomial function of the derivative \(\frac{d^2y}{dx^2}\). Since sine is a transcendental function, the equation cannot be expressed as a polynomial in derivatives.

**Hint:** Check if any of the terms involve non-polynomial functions of the derivatives.

### Step 4: Conclude the degree of the differential equation
Since the equation contains a non-polynomial term involving the derivative, the degree of the differential equation is not defined.

**Hint:** If the equation contains non-polynomial expressions involving derivatives, the degree is considered not defined.

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