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lecture_16 (dragged) - MA 36600 LECTURE NOTES: MONDAY,...

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MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 23 3 is nonzero then y ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t ) is the solution of the initial value problem in terms of the constants c 1 = y 0 y ° 2 ( t 0 ) y ° 0 y 2 ( t 0 ) W 0 and c 2 = y 0 y ° 1 ( t 0 ) y ° 0 y 1 ( t 0 ) W 0 . But we saw above that W ° y 1 ,y 2 ± ( t 0 )= W 0 ° = 0 if and only if y 1 and y 2 are linearly independent. In particular, to check that two solutions y 1 = y 1 ( t ) and y 2 = y 2 ( t ) form the general solution, it suffices to check that their Wronskian W ° y 1 ,y 2 ± ( t )= y 1 ( t ) y ° 2 ( t ) y ° 1 ( t ) y 2 ( t ) is nonzero for some t = t 0 . In this case, we say that { y 1 ,y 2 } forms a Fundamental Set of Solutions to the diFerential equation. We have seen that if y 1 = y 1 ( t ) and y 2 = y 2 ( t ) are linearly independent solutions to some diFerential equation in the form a ( t ) y °° + b ( t ) y ° + c ( t ) y =0 then we can ±nd the general solution y ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t ). We mention in passing that given two linearly independent functions f = f ( t ) and g = g ( t ), we can always ±nd a homogeneous second order diFerential equation y °° + p
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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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