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CHARACTERISTIC EQUATIONS
Methods for determining the roots, characteristic equation and general solution
used in solving second order constant coefficient differential equations
There are three types of roots, Distinct, Repeated and Complex, which determine which of the three types
of general solutions is used in solving a problem.
Distinct Real Roots
If the roots have opposite sign,
the graph will be have a
saddle
point
where only two asymptotic
curves intersect.
If the roots are
unequal with the same sign, there
are many curves intersecting at a
critical point.
Repeated Roots
If the roots are real and equal,
the graph of the equation will
have
multiple curves
that
intersect at a critical point.
Complex Roots
If the roots are pure imaginary, the
graph will have
circles or ovals
around
a critical point.
If the roots are complex
conjugates, the graph will have a critical
point anchoring a
spiral
.
If the real
part of the conjugate is positive, the
spiral is expanding (direction of
movement is outward); negative means
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 Spring '08
 comech
 Differential Equations, Equations

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