Question

If \( \sin x \) is a solution of the differential equation \[ \dfrac{d^4 y}{dx^4} + 2 \dfrac{d^3 y}{dx^3} + 6 \dfrac{d^2 y}{dx^2} + 2 \dfrac{dy}{dx} + 5y = 0, \] then the general solution is ...........

• A. \( y = C_1 \sin x + e^{-x}(C_2 \sin 2x + C_3 \cos 2x) \)
• B. \( y = C_1 \sin x + C_2 \cos x + e^{-x}(C_3 \sin 2x + C_4 \cos 2x) \)
• C. \( y = C_1 \sin x + C_2 \cos x + C_3 \sin 2x + C_4 \cos 2x \)
• D. \( y = C_1 \sin x + C_2 \cos x + C_3 e^{-3x} + C_4 e^{-2x} \)

Hint:

When given a higher-order linear differential equation with constant coefficients, try substituting \( y = e^{mx} \) to form the characteristic equation and solve for the roots to build the general solution.

Answer 1

The Correct Option is B

Step 1: Assume the solution is of the form \( y = e^{mx} \) and substitute into the equation:
\[ m^4 + 2m^3 + 6m^2 + 2m + 5 = 0 \] Use substitution or trial to factor this quartic. It factors into complex roots: \[ m = \pm i, \quad m = -1 \pm 2i \] Step 2: General solution using roots:
- \( m = i ⇒ \sin x, \cos x \)
- \( m = -1 \pm 2i ⇒ e^{-x} \sin 2x, e^{-x} \cos 2x \)
So, the general solution is:
\[ y = C_1 \sin x + C_2 \cos x + e^{-x}(C_3 \sin 2x + C_4 \cos 2x) \] Final Answer: \( \boxed{y = C_1 \sin x + C_2 \cos x + e^{-x}(C_3 \sin 2x + C_4 \cos 2x)} \)