Order and degree of the differential equation `(d^(2)y)/(dx^(2))+x(dy)/(dx)+y=2sinx` respectively are

Options

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

Answer

To find the order and degree of the given differential equation:

\[ \frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 2\sin x \]

we will follow these steps:

### Step 1: Identify the highest derivative
The first step is to identify the highest derivative present in the equation.

In the given equation, we have:
- The second derivative \(\frac{d^2y}{dx^2}\)
- The first derivative \(\frac{dy}{dx}\)

The highest derivative here is \(\frac{d^2y}{dx^2}\).

### Step 2: Determine the order
The order of a differential equation is defined as the highest derivative present in the equation.

Since the highest derivative is \(\frac{d^2y}{dx^2}\), the order of the differential equation is:
\[ \text{Order} = 2 \]

### Step 3: Determine the degree
The degree of a differential equation is defined as the power of the highest derivative when the equation is expressed in polynomial form.

In this case, the highest derivative \(\frac{d^2y}{dx^2}\) appears to the power of 1 (it is not under a root or raised to any power other than 1). Therefore, the degree of the differential equation is:
\[ \text{Degree} = 1 \]

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

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