The general solution to the second order linear homogeneous differential equation y″ − 6y′ + 25y = 0 is
Options:
(A) e³ˣ(a cos 4x + b sin 4x)
(B) e³ⁱˣ(a cos 4x + b sin 4x)
(C) e⁴ˣ(a cos 3x + b sin 3x)
(D) e⁴ⁱˣ(a cos 3x + b sin 3x)
Official Answer Key:
A
Step-by-Step Solution & Explanation
Auxiliary equation: m² − 6m + 25 = 0.
m = [6 ± √(36 − 100)]/2 = [6 ± √(−64)]/2 = 3 ± 4i.
For complex roots α ± iβ the solution is e^{αx}(a cos βx + b sin βx), with α = 3, β = 4.
y = e³ˣ(a cos 4x + b sin 4x).
Distractor (C) swaps the real and imaginary parts of the root.
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