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Unformatted text preview: Notes for Day : . : Rea Repeated Roots/Reduction of Order Review Last time, we discussed the characteristic polynomial ak + bk + c associated with the linear homogeneous equation ay + by + cy = . We showed that when the characteristic polynomial has two distinct real roots k and k , then e k t and e k t form a fundamental set of solutions, and so the general solution must be y = c e k t + c e k t . We also mentioned that when the characteristic polynomial has only one root, k , the general solution has the alternate form y = ( c t + c ) e k t . Today, well prove that fact using a technique called reduction of order , and well show how to nd the general solution of the di erential equation when the characteristic polynomial has no real roots. Reduction of Order Suppose we had a second order linear homogeneous di erential equation y + p ( t ) y + q ( t ) y = , ( ) in which q ( t ) = , so that we have: y + p ( t ) y = ( ) Despite the appearances, this last equation is actually a rst-order di erential equation, if you take the original function to be...
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This note was uploaded on 04/05/2008 for the course MATH 2214 taught by Professor Edesturler during the Spring '06 term at Virginia Tech.
- Spring '06