Consider the following in respect of the differential equation `(d^(2)y)/(dx^(2)) + 2 ((dy)/(dx)) + 9y = x`
1. degree of the differential equation is 1
2. order of the differential equation is 2
Which of the above statement is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the problem, we need to analyze the given differential equation:

\[ \frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 9y = x \]

### Step 1: 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 equation, the highest derivative is \(\frac{d^2y}{dx^2}\), which is a second derivative.

**Conclusion for Step 1:** The order of the differential equation is 2.

### Step 2: Identify the Degree of the Differential Equation
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made a polynomial in derivatives.

- In our equation, the highest order derivative is \(\frac{d^2y}{dx^2}\) and it appears to the power of 1.

**Conclusion for Step 2:** The degree of the differential equation is 1.

### Final Conclusion
1. The degree of the differential equation is 1 (True).
2. The order of the differential equation is 2 (True).

Both statements are correct.

### Answer
The correct option is C: Both 1 and 2 are correct.

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