Order and degree of the differential equation `(dy)/(dx)=(2sinx+3)/((dy)/(dx))` respectively are

Options

A. 1 and 1
B. 1 and 2
C. 2 and 1
D. 2 and 2

Answer

To find the order and degree of the given differential equation \(\frac{dy}{dx} = \frac{2\sin x + 3}{\frac{dy}{dx}}\), we will follow these steps:

### Step 1: Rewrite the differential equation
We start with the given equation:
\[ \frac{dy}{dx} = \frac{2\sin x + 3}{\frac{dy}{dx}} \]
To eliminate the fraction, we can multiply both sides by \(\frac{dy}{dx}\):
\[ \left(\frac{dy}{dx}\right)^2 = 2\sin x + 3 \]

### Step 2: Identify the order of the differential equation
The order of a differential equation is determined by the highest derivative present in the equation. In our case, we have:
\[ \left(\frac{dy}{dx}\right)^2 = 2\sin x + 3 \]
The highest derivative is \(\frac{dy}{dx}\), which is a first-order derivative. Therefore, the order of the differential equation is:
\[ \text{Order} = 1 \]

### Step 3: Identify the degree of the differential equation
The degree of a differential equation is determined by the highest power of the highest derivative when the equation is expressed as a polynomial in derivatives. In our case, we have:
\[ \left(\frac{dy}{dx}\right)^2 = 2\sin x + 3 \]
The highest derivative is \(\frac{dy}{dx}\) and its power in this equation is 2. Therefore, the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Final Answer
Thus, the order and degree of the differential equation are:
- **Order**: 1
- **Degree**: 2

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