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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. Complex-valued solutions Up to this point we have been working over the real numbers, looking for real-valued 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 real-valued 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 real-valued 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 higher-order derivatives. Then if y ( x ) = u ( x ) + iv ( x ) is complex-valued, 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.