Question

The solution of the second-order differential equation \[ \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = 0 \] with boundary conditions \( y(0) = 1 \) and \( y(1) = 3 \) is

• A. \( e^{-x} + (3e - 1) x e^{-x} \)
• B. \( e^{-x} - (3e - 1) x e^{-x} \)
• C. \( e^{-x} + \left[ 3 \sin\left( \frac{\pi x}{2} \right) - 1 \right] x e^{-x} \)
• D. \( e^{-x} - \left[ 3 \sin\left( \frac{\pi x}{2} \right) - 1 \right] x e^{-x} \)

Hint:

For second-order linear differential equations with constant coefficients, solve the characteristic equation to find the general solution. Then, apply the boundary conditions to determine the constants.

Answer 1

The Correct Option is A

The given second-order differential equation is: \[ \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = 0 \] The characteristic equation corresponding to this differential equation is: \[ r^2 + 2r + 1 = 0 \] Solving for \( r \), we get a double root \( r = -1 \). Therefore, the general solution to the differential equation is: \[ y(x) = (A + Bx) e^{-x} \] Using the given boundary conditions \( y(0) = 1 \) and \( y(1) = 3 \): - For \( y(0) = 1 \): \[ A = 1 \] - For \( y(1) = 3 \): \[ 1 + B e^{-1} = 3 $\Rightarrow$ B = (3 - 1) e = 2e \] Thus, the solution is: \[ y(x) = \left( 1 + 2e x \right) e^{-x} \] which simplifies to: \[ y(x) = e^{-x} + (3e - 1) x e^{-x} \] Therefore, the correct answer is option (A).
Final Answer: (A) \( e^{-x} + (3e - 1) x e^{-x} \)