c12 - 18.03 Class 12, March 6, 2006 Homogeneous constant...

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18.03 Class 12, March 6, 2006 Homogeneous constant coefficient linear equations: complex or repeated roots, damping criteria. [1] We are studying equations of the form x" + b x' + k x = 0 (*) which model a mass, dashpot, spring system without external forcing term. We found that (*) has an exponential solution e^{rt} exactly when r is a root of the "characteristic polynomial" p(s) = s^2 + bs + k Example A. x" + 5x' + 4x = 0 . We did this: The characteristic polynomial s^2 + 5s + 4 factors as (s + 1)(s + 4) so the roots are r = -1 and r = -4 . The corresponding exponential solutions are e^{-t} and e^{-4t} . The general solution is a linear combination of these: x = c1 e^{-t} + c2 e^{-4t} . All solutions go to zero: no oscillation here. When the roots are real and not equal to each other the system is called "Overdamped." Example B. x" + 4x' + 5x = 0 The characteristic polynomial s^2 + 4s + 5 has roots r = -2 +- sqrt(4-5) = -2 +- i Our old friend i = sqrt(-1) appears, and we have exponential solutions e^{(-2+i)t} , e^{(-2-i)t} I guess we were expecting REAL valued solutions. For this we have: Theorem: If x is a complex-valued solution to mx" + bx' + kx = 0, where m, b, and k are real, then the real and imaginary parts of x are also solutions. Proof:
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c12 - 18.03 Class 12, March 6, 2006 Homogeneous constant...

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