General solution to the differential equation y″ + 4y′ + 5y = 0 is
Options:
(A) e^{2x}
(B) e^{-2x}
(C) e^{x}
(D) e^{-x}
Official Answer Key:
B
Step-by-Step Solution & Explanation
Auxiliary equation: r² + 4r + 5 = 0. r = [−4 ± √(16−20)]/2 = [−4 ± √(−4)]/2 = −2 ± i. Complex roots α ± iβ with α = −2, β = 1 give the general solution y = e^{αx}(a cos βx + b sin βx) = e^{-2x}(a cos x + b sin x). (A) has the wrong sign on the exponent; (C), (D) have the wrong imaginary part (β=2 instead of 1). Answer: (B).
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