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2 MA 36600 LECTURE NOTES: MONDAY, MARCH 30 (3) Y = Y ( t ) is a solution to the nonhomogeneous equation n j =0 P n j ( t ) Y ( j ) = G ( t ) . Then the general solution to the di ff erential equation is y ( t ) = y c ( t ) + Y ( t ) where y c ( t ) = n i =1 C i y i ( t ) . For example, consider now the constant coe cient equation a 0 d n y dt n + a 1 d n 1 y dt n 1 + · · · + a n 1 dy dt + a n y = G ( t ) . Say that we can factor Z ( r ) = n j =0 a n j r j = a 0 · p k =1 r r k s k real roots · q k =1 r [ λ k + i μ k ] r [ λ k i μ k ] s k complex roots . Then the general solution is y ( t ) = y c ( t ) + Y ( t ), where y c ( t ) = p k =1 s k m =1 C km t m 1 e r k t + q k =1 s k m =1 A km t m 1 e λ k t cos μ k t + s k m =1 B km t m 1 e λ k t sin μ k t . We discuss a couple of ways to compute Y = Y ( t ). Method of Undetermined Coe cients. Consider the nonhomogeneous di ff erential equation a 0 d n y dt n + a 1 d n 1 y dt n 1 + · · · + a n 1 dy dt + a n y = G ( t ) . We outline a method to determine a particular solution Y = Y ( t ). #1. Express the function on the right-hand side as the sum of functions G ( t ) = G 1 ( t ) + G 2 ( t ) + · · · + G r ( t ) where each G i ( t ) is the product of a polynomial, an exponential function, and a trigonometric function. That is, say that we can write G i ( t ) = d i j =0 a ij t j e α i t cos β i t + d i j =0 b ij t j e α i t sin β i t for some constants
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