Question

The solution of the differential equation \( \frac{d^2 y}{dx^2} + y = 0 \) with \( y(0) = 1, y\left( \frac{\pi}{2} \right) = 2 \) is

• A. \( y = 2 \cos x + \sin x \)
• B. \( y = \cos x + 2 \sin x \)
• C. \( y = \cos x + \sin x \)
• D. \( y = 2(\cos x + \sin x) \)

Hint:

For second-order homogeneous differential equations with constant coefficients, the general solution often involves sine and cosine functions, with the coefficients determined by the initial conditions.

Answer 1

The Correct Option is D

Step 1: Understanding the differential equation.
The given differential equation is: \[ \frac{d^2 y}{dx^2} + y = 0 \] This is a second-order linear homogeneous differential equation with constant coefficients. The general form of the solution to this equation is: \[ y(x) = C_1 \cos x + C_2 \sin x \] This solution comes from the fact that the characteristic equation for this differential equation is \( r^2 + 1 = 0 \), which has the roots \( r = \pm i \), leading to a solution involving sine and cosine functions.

Step 2: Applying the initial conditions.
We are given the initial conditions: 1. \( y(0) = 1 \) 2. \( y\left( \frac{\pi}{2} \right) = 2 \) Let’s first apply the initial condition \( y(0) = 1 \): Substitute \( x = 0 \) into the general solution: \[ y(0) = C_1 \cos 0 + C_2 \sin 0 = C_1 \] Since \( y(0) = 1 \), we get: \[ C_1 = 1 \] Now, apply the second initial condition \( y\left( \frac{\pi}{2} \right) = 2 \): Substitute \( x = \frac{\pi}{2} \) into the general solution: \[ y\left( \frac{\pi}{2} \right) = C_1 \cos \frac{\pi}{2} + C_2 \sin \frac{\pi}{2} \] Since \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \), this simplifies to: \[ y\left( \frac{\pi}{2} \right) = C_2 \] Since \( y\left( \frac{\pi}{2} \right) = 2 \), we have: \[ C_2 = 2 \]
Step 3: Final solution.
Substitute the values of \( C_1 \) and \( C_2 \) into the general solution: \[ y(x) = 1 \cdot \cos x + 2 \cdot \sin x = \cos x + 2 \sin x \] Thus, the solution is: \[ y(x) = 2 (\cos x + \sin x) \]