SECOND ORDER, LINEAR, HOMOGENEOUS O.D.E.S AS AN EXAMPLE OF THIS CLASS:

ay'' + by' + cy = 0

USE THE TRIAL SOLUTION y = e?x

THEN THE DERIVATIVES ARE:

y' = ?e?x y'' = ?2e?x

SUBSTITUTING IN THE DIFFERENTIAL EQUATION:

a ?2e?x + b ?e?x + c e?x = 0

MULTIPLYING BY e-?x

a ?2 + b ? + c = 0

THIS IS THE CHARACTERISTIC EQUATION

ITS ROOTS DETERMINE THE VALUUES

OF ? IN THE TRIAL SOLUTIONS

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