What is the order of the differential equation `((dy)/(dx))^(2) + (dy)/(dx) - sin^(2) y = 0` ?

Options

A. 1
B. 2
C. 3
D. Undefined

Answer

To determine the order of the differential equation \(\left(\frac{dy}{dx}\right)^{2} + \frac{dy}{dx} - \sin^{2} y = 0\), we will follow these steps:

### Step 1: Identify the derivatives present in the equation
In the given equation, we have the term \(\frac{dy}{dx}\) which represents the first derivative of \(y\) with respect to \(x\).

### Step 2: Determine the highest order derivative
The highest order derivative present in the equation is \(\frac{dy}{dx}\). There are no higher derivatives such as \(\frac{d^2y}{dx^2}\) (the second derivative) in the equation.

### Step 3: Conclude the order of the differential equation
Since the highest derivative present is the first derivative \(\frac{dy}{dx}\), the order of the differential equation is 1.

### Final Answer
The order of the differential equation \(\left(\frac{dy}{dx}\right)^{2} + \frac{dy}{dx} - \sin^{2} y = 0\) is **1**.

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