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ch03_5 - Ch 3.5 Repeated Roots Reduction of Order Recall...

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Ch 3.5: Repeated Roots; Reduction of Order Recall our 2 nd order linear homogeneous ODE 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 : When b 2 – 4 ac = 0, r 1 = r 2 = - b /2 a , since method only gives one solution: 0 = + + cy y b y a 0 ) ( 2 = + + = c br ar e t y rt a ac b b r 2 4 2 - ± - = a t b ce t y 2 / 1 ) ( - =
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Second Solution: Multiplying Factor v ( t ) We know that Since y 1 and y 2 are linearly dependent, we generalize this approach and multiply by a function v , and determine conditions for which y 2 is a solution: Then solution a ) ( ) ( solution a ) ( 1 2 1 t cy t y t y = a t b a t b e t v t y e t y 2 / 2 2 / 1 ) ( ) ( try solution a ) ( - - = = a t b a t b a t b a t b a t b a t b a t b e t v a b e t v a b e t v a b e t v t y e t v a b e t v t y e t v t y 2 / 2 2 2 / 2 / 2 / 2 2 / 2 / 2 2 / 2 ) ( 4 ) ( 2 ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( - - - - - - - + - - = - = =
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Finding Multiplying Factor v ( t ) Substituting derivatives into ODE, we seek a formula for v : 0 = + + cy y b y a 4 3 2 2 2 2 2 2 2 2 2 2 2 / ) ( 0 ) ( 0 ) ( 4 4 ) ( 0 ) ( 4 4 4 ) ( 0 ) ( 4 4 4 2 4 ) ( 0 ) ( 2 4 ) ( 0 ) ( ) ( 2 ) ( ) ( 4 ) ( ) ( 0 ) ( ) ( 2 ) ( ) ( 4 ) ( ) ( k t k t v t v t v a ac b t v a t v a ac a b t v a t v a ac a b a b t v a t v c a b a b t v a t cv t v a b t v b t v a b t v b t v a t cv t v a b t v b t v a b t v a b t v a e
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