lecture_17 (dragged) 2

lecture_17 (dragged) 2 - MA 36600 LECTURE NOTES WEDNESDAY...

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MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 25 3 Repeated Roots Review. We recap what we know so far. Consider the constant coeﬃcient diFerential equation ay °° + by ° + cy =0 We know that we can ±nd solutions by considering the characteristic equation ar 2 + br + c =0 . Say that roots are r 1 and r 2 . Using the Quadratic ²ormula, we can explicitly write r 1 = b 2 a + b 2 4 ac 2 a and r 2 = b 2 a b 2 4 ac 2 a . If the discriminant b 2 4 ac> 0, we have seen that a fundamental set of solutions is given by y 1 ( t )= e r 1 t and y 2 ( t )= e r 2 t . If the discriminant b 2 4 ac< 0, w have seen that a fundamental set of solutions is given by y 1 ( t )= e λt cos μt and y 2 ( t )= e λt sin μt. What if the discriminant b 2 4 ac = 0 i.e., we have repeated roots? How do we ±nd all solutions? We consider an example for motivation. Consider the diFerential equation y °° =0 . Then the characteristic equation r 2 = 0 has the roots r 1 = r 2 = 0. We know that one solution if y 1 ( t )= e r 1 t = 1. We try a diFerent method in order to ±nd all solutions to the equation. We see that
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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