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Unformatted text preview: Linear Eq’s With Constant Coefficients We continue to look at the differential equation ay 00 + by + cy = 0 but now consider the case where the roots of the characteristic equation are complex. We first build some tools to help us toward this goal. Complexvalued solutions Up to this point we have been working over the real numbers, looking for realvalued solutions to our differential equations. As is often the case in mathematics, we must sometimes enlarge our perspective temporarily in order to solve a specific problem. Here, we will temporarily work over the larger field of complex numbers in order to find two realvalued solutions to our differential equation. We begin with some preliminaries. Notice that if y ( x ) is a function from the real numbers to the complex numbers, it must have the form y ( x ) = u ( x ) + iv ( x ) where u ( x ) and v ( x ) are realvalued functions. Here u ( x ) is called the real part of y ( x ) and v ( x ) is called the imaginary part. We can the derivatives of such functions by y ( x ) = y ( x ) + iv ( x ) and similarly for higherorder derivatives. Then if y ( x ) = u ( x ) + iv ( x ) is complexvalued, we have ay 00 + by + cy = 0 m a ( u 00 + iv 00 ) + b ( u + iv ) + c...
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This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.
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