ch03_4 - Ch 3.4: Complex Roots of Characteristic Equation...

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Unformatted text preview: Ch 3.4: Complex Roots of Characteristic Equation Recall our discussion of the equation where a , b and c are constants. Assuming an exponential soln leads to characteristic equation: Quadratic formula (or factoring) yields two solutions, r 1 & r 2 : If b 2 4 ac < 0, then complex roots: r 1 = + i , r 2 = - i Thus = + + cy y b y a ) ( 2 = + + = c br ar e t y rt a ac b b r 2 4 2- - = ( 29 ( 29 t i t i e t y e t y - + = = ) ( , ) ( 2 1 Eulers Formula; Complex Valued Solutions Substituting it into Taylor series for e t , we obtain Eulers formula : Generalizing Eulers formula, we obtain Then Therefore ( 29 ( 29 t i t n t i n t n it e n n n n n n n n it sin cos ! 1 2 ) 1 ( ! 2 ) 1 ( ! ) ( 1 1 2 1 2 + =-- +- = = =-- = = t i t e t i sin cos + = ( 29 [ ] t ie t e t i t e e e e t t t t i t t i sin cos sin cos + = + = = + ( 29 ( 29 t ie t e e t y t ie t e e t y t t t i t t t i...
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ch03_4 - Ch 3.4: Complex Roots of Characteristic Equation...

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