Step 1: Understanding the Concept:
The order of a differential equation is fundamentally linked to the number of arbitrary constants in its general solution. To find the differential equation from a general solution, we must differentiate the solution enough times to be able to eliminate all the arbitrary constants.
Step 2: Key Formula or Approach:
The order of a differential equation is equal to the number of essential arbitrary constants in its general solution.
Step 3: Detailed Explanation:
The given general solution is:
\[ y = a \sin x + b \cos x \]
This solution contains two arbitrary constants, 'a' and 'b'.
Therefore, the order of the corresponding differential equation must be 2.
To verify this, we can form the differential equation:
Differentiate the solution with respect to x:
\[ \frac{dy}{dx} = a \cos x - b \sin x \quad \dots(1) \]
Differentiate again with respect to x:
\[ \frac{d^2y}{dx^2} = -a \sin x - b \cos x \quad \dots(2) \]
We can see that the right-hand side of equation (2) is the negative of the original expression for y.
\[ \frac{d^2y}{dx^2} = -(a \sin x + b \cos x) \]
\[ \frac{d^2y}{dx^2} = -y \]
Rearranging this gives the differential equation:
\[ \frac{d^2y}{dx^2} + y = 0 \]
This is a second-order differential equation, as predicted.
Step 4: Final Answer:
Since there are two arbitrary constants in the general solution, the order of the differential equation is 2. Therefore, option (A) is correct.