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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