Question

The order of the differential equation whose general solution is \(y = a \sin x + b \cos x\) is (where a and b are arbitrary constants)

• A. 2
• B. 4
• C. 1
• D. 3

Hint:

A very quick way to solve this type of problem is to simply count the number of independent arbitrary constants in the general solution. This count directly gives you the order of the differential equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the order of the differential equation that corresponds to the given general solution.

Step 2: Key Formula or Approach:
The order of a differential equation is equal to the number of independent arbitrary constants in its general solution. To form the differential equation, we need to differentiate the solution as many times as there are constants and then eliminate the constants.

Step 3: Detailed Explanation:
The given general solution is:
\[ y = a \sin x + b \cos x \]
This solution contains two independent arbitrary constants, \(a\) and \(b\).
According to the rule, the order of the differential equation must be equal to the number of these constants.
Therefore, the order of the differential equation is 2.

Derivation (for verification):
1. Differentiate the solution with respect to \(x\):
\[ \frac{dy}{dx} = a \cos x - b \sin x \]
2. Differentiate a second time to eliminate the constants:
\[ \frac{d^2y}{dx^2} = -a \sin x - b \cos x \]
Notice that the right side is the negative of the original expression for \(y\).
\[ \frac{d^2y}{dx^2} = -(a \sin x + b \cos x) = -y \]
So, the differential equation is:
\[ \frac{d^2y}{dx^2} + y = 0 \]
The highest order derivative in this equation is the second derivative, so the order is 2. This confirms our initial conclusion.

Step 4: Final Answer:
The order of the differential equation is 2.