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

Options

A. 1
B. 2
C. 3
D. not defined

Answer

To determine the degree of the given differential equation

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

we will follow these steps:

### Step 1: Identify the highest order derivative
The highest order derivative in the equation is \(\frac{d^2y}{dx^2}\), which is the second derivative of \(y\) with respect to \(x\).

### Step 2: Check if the equation is a polynomial in derivatives
The degree of a differential equation is defined only if the equation can be expressed as a polynomial in its derivatives. In our case, we have:

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

The left-hand side is a polynomial in the derivatives \(\frac{d^2y}{dx^2}\) and \(\frac{dy}{dx}\).

### Step 3: Analyze the right-hand side
The right-hand side, \(\sin\left(\frac{dy}{dx}\right)\), is not a polynomial in \(\frac{dy}{dx}\). The sine function introduces a non-polynomial term, which complicates the definition of the degree.

### Step 4: Conclusion about the degree
Since the right-hand side includes a sine function of a derivative, the overall equation cannot be classified as a polynomial in its derivatives. Therefore, the degree of the differential equation is not defined.

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

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