Economics Dynamics Problems 78

Economics Dynamics Problems 78 - (0) ry (0)] te rt Example...

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62 Economic Dynamics If y (0) = 0 and y ± (0) = 1, then c 1 = 1 5 1 =− 1 6 c 2 = 1 1 ( 5) = 1 6 So the particular solution is y ( t ) =− ± 1 6 ² e 5 t + ± 1 6 ² e t 2.8.2 Real and equal roots ( b 2 = 4 ac ) If r is a repeated real root to the differential equation ay ±± ( t ) + by ± ( t ) + c = 0 then a general solution is y ( t ) = c 1 e rt + c 2 te rt where c 1 and c 2 are arbitrary constants (see exercise 9). If y (0) and y ± (0) are the two initial conditions, then y (0) = c 1 + c 2 (0) = c 1 y ± ( t ) = rc 1 e rt + rc 2 te rt + c 2 e rt y ± (0) = rc 1 + c 2 Hence c 1 = y (0) , c 2 = y ± (0) ry (0) So the particular solution is y ( t ) = y (0) e rt + [ y ±
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Unformatted text preview: (0) ry (0)] te rt Example 2.18 y ( t ) + 4 y ( t ) + 4 y ( t ) = Then the auxiliary equation is x 2 + 4 x + 4 = ( x + 2) 2 = Hence, r = 2 and the general solution is y ( t ) = c 1 e 2 t + c 2 te 2 t If y (0) = 3 and y (0) = 7, then c 1 = y (0) = 3 c 2 = y (0) ry (0) = 7 ( 2)(3) = 13...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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