L43 - Hand Examples Fall 2003 Math 308/501502 4 Linear...

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Fall 2003 Math 308/501–502 4 Linear Second-Order Equations 4.3 Auxiliary Eqs with Complex Roots Mon, 29/Sep c ± 2003, Art Belmonte Summary Terminology and Solutions Consider the ODE ay 00 + by 0 + cy = 0, where a , b , c are constants (with a 6= 0) and t is the independent variable (for specificity). The ODE has an associated auxiliary or characteristic equation ar 2 + br + c = 0. This quadratic equation has roots r = r 1 , r 2 = - b ² p b 2 - 4 ac 2 a , the discriminant of which is b 2 - 4 . In the following , c 1 and c 2 are arbitrary constants. When b 2 - 4 > 0, the roots are real and distinct. In this case, y 1 ( t ) = e r 1 t and y 2 ( t ) = e r 2 t form a fundamental set of solutions to the ODE. A general solution is y ( t ) = c 1 e r 1 t + c 2 e r 2 t . When b 2 - 4 = 0, there is a repeated or double root. In this case y 1 ( t ) = e rt and y 2 ( t ) = te form a fundamental set of solutions to the ODE. A general solution is y ( t ) = c 1 e + c 2 . When b 2 - 4 < 0, the roots are complex conjugate numbers, r = α ² β i ,where α =- b 2 a , β = b 2 - 4 2 a ,and i = - 1.Here y 1 ( t ) = e α t cos β t and y 2 ( t ) = e α t sin β t form a (real) fundamental set of solutions to the ODE.
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This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L43 - Hand Examples Fall 2003 Math 308/501502 4 Linear...

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