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: Write the given differential equation. 
The equation is: \[ y'' + y = 0 \]

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

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

Step 4: Check each option. 
- (A) $\cos x$: Clearly a solution, since it fits the general solution. 
- (B) $\sin x$: Also a solution, since it fits the general solution. 
- (C) $1 + \cos x$: Not a solution, because the constant term $1$ does not satisfy the equation. 
- (D) $1 + \sin x$: Not a solution, because the constant term $1$ does not satisfy the equation. 
 

Step 5: Conclusion. 
Thus, the correct solutions are (A) and (B), making the correct answer (2) A and B only.