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

Options

A. 1
B. 2
C. 3
D. Nor defined

Answer

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

### Step 2: Check the nature of the differential equation
For a differential equation to have a defined degree, it must be polynomial in terms of its derivatives. This means that the highest order derivative should not be involved in any transcendental functions (like sine, cosine, exponential, etc.).

### Step 3: Analyze the equation
In our equation, the term \(\sin\left(\frac{d^2y}{dx^2}\right)\) involves the highest order derivative \(\frac{d^2y}{dx^2}\) in a sine function. This is a non-polynomial form.

### Step 4: Conclusion about the degree
Since the highest order derivative \(\frac{d^2y}{dx^2}\) is present inside a sine function, the differential equation does not satisfy the polynomial condition. Therefore, the degree of the differential equation is not defined.

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

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