Question

The solution(s) of the ordinary differential equation $y'' + y = 0$, is: 

(A) $\cos x$ 
(B) $\sin x$ 
(C) $1 + \cos x$ 
(D) $1 + \sin x$ 
Choose the most appropriate answer from the options given below:

• A. A and D only
• B. A and B only
• C. C and D only
• D. B and C only

Hint:

For linear differential equations with constant coefficients, solve using the auxiliary equation method to find the general solution.

Answer 1

The Correct Option is B

Step 1: State the differential equation.
The equation is: \[ y'' + y = 0 \]

Step 2: Determine the auxiliary equation.
The auxiliary equation is: \[ m^2 + 1 = 0 \Rightarrow m = \pm i \]

Step 3: Formulate the general solution.
The general solution to the differential equation is: \[ y(x) = C_1 \cos x + C_2 \sin x \]

Step 4: Evaluate each option.
- (A) $\cos x$: This is a solution as it aligns with the general solution.
- (B) $\sin x$: This is also a solution, fitting the general solution.
- (C) $1 + \cos x$: This is not a solution, as the constant term $1$ does not satisfy the equation.
- (D) $1 + \sin x$: This is not a solution, due to the constant term $1$ not satisfying the equation.

Step 5: Conclude.
Therefore, options (A) and (B) are the correct solutions, indicating the correct answer is (2) A and B only.