Continuous dynamic systems 65 (3) Complex conjugate roots y c = c 1 e α t cos( β t ) + c 2 e α t sin( β t ) In fnding a solution to a linear nonhomogeneous equation, Four steps need to be Followed: Step 1 ±ind the complementary solution y c . Step 2 ±ind the general solution y h by solving the higher-order equation L h ( y h ) =0 where y h is determined From L ( y ) and g ( t ). Step 3 Obtain y q = y h − y c . Step 4 Determine the unknown constant, the undetermined coefFcients ,inthe solution y q by requiring L ( y q ) = g ( t ) and substituting these into y q , giving the particular solution y p . Example 2.20 Suppose y ±± ( t ) + y ± ( t ) = t Step 1 This has the complementary solution y c , which is the solution to the aux-iliary equation x 2 + x =0 x ( x + 1) =0 with solutions r = 0 and s =− 1 and
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