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lecture_17 (dragged) 3

# lecture_17 (dragged) 3 - y 1 t = e rt and y 2 t = t e rt...

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4 MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 25 In particular, ay °° + by ° + cy = 0 if and only if v °° = 0. But we know that the general solution to this diFerential equation is v ( t )= c 1 t + c 2 for constants c 1 and c 2 . Hence the general solution to the diFerential equation ay °° + by ° + cy =0isthe function y ( t )= c 1 te rt + c 2 e rt . This technique of ±nding the general solution when b 2 4 ac = 0 is called d’Alembert’s Method . We remark in passing that the functions
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Unformatted text preview: y 1 ( t ) = e rt and y 2 ( t ) = t e rt are linearly independent. Indeed, the Wronskian of these two functions is W ° y 1 , y 2 ± ( t ) = y 1 ( t ) y ° 2 ( t ) − y ° 1 ( t ) y 2 ( t ) = e rt which is a nonzero function. Hence { y 1 , y 2 } forms a fundamental set of solutions for the diFerential equation....
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