If `y = A sin (theta + B)`, where A and B are arbitrary constant then to form a differential equation how many times it should be differentiated ?

Options

A. 1
B. 2
C. 3
D. Cannot be formed

Answer

To form a differential equation from the given function \( y = A \sin(\theta + B) \), where \( A \) and \( B \) are arbitrary constants, we need to determine how many times we should differentiate the function.

### Step-by-Step Solution:

1. **Identify the Function**:
The given function is:
\[ y = A \sin(\theta + B) \]
Here, \( A \) and \( B \) are constants.

2. **Count the Constants**:
In the equation, we have two arbitrary constants: \( A \) and \( B \).

3. **Differentiate the Function**:
To eliminate the constants and form a differential equation, we need to differentiate the function. Each differentiation can potentially eliminate one constant.

4. **First Differentiation**:
Differentiate \( y \) with respect to \( \theta \):
\[ \frac{dy}{d\theta} = A \cos(\theta + B) \]
At this stage, we still have one constant \( A \) remaining.

5. **Second Differentiation**:
Differentiate again:
\[ \frac{d^2y}{d\theta^2} = -A \sin(\theta + B) \]
Now, both constants \( A \) and \( B \) have been eliminated.

6. **Conclusion**:
Since we have two arbitrary constants, we need to differentiate the original equation **twice** to form a differential equation.

Thus, the answer is that we need to differentiate the function **2 times**.