lecture_39

# lecture_39 - MA 36600 LECTURE NOTES WEDNESDAY APRIL 29...

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MA 36600 LECTURE NOTES: WEDNESDAY, APRIL 29 Nonhomogeneous Systems Constant Coeﬃcients. Consider now the nonhomogeneous system d dt x = Ax + g ( t ) where A is a constant n × n matrix. We will show that the general solution to this system is in the form x ( t ) = x ( c ) ( t ) + x ( p ) ( t ) in terms of a homogeneous solution x ( c ) ( t ) = Ψ ( t ) c , where Ψ ( t ) = exp ( A t ); and a particular solution x ( p ) ( t ) = Ψ ( t ) Z t Ψ ( τ ) - 1 g ( τ ) dτ. We use integrating factors, just like before. Recall the Product Rule for Matrices: d dt x = Ax + g ( t ) d dt x - Ax = g ( t ) exp ( - A t ) · d dt x + ± - A exp ( - A t ) ² · x = exp ( - A t ) g ( t ) d dt ³ exp ( - A t ) · x ´ = exp ( - A t ) g ( t ) exp ( - A t ) · x ( t ) = c + Z t exp ( - A τ ) g ( τ ) for some constant vector c . Now use the fact that Ψ ( t ) = exp ( A t ) = Ψ ( t ) - 1 = exp ( - A t ) . Then we have the expression Ψ ( t ) - 1 · x ( t ) = c + Z t Ψ ( τ ) - 1 g ( τ ) dτ. The claim follows when we multiply both sides by the fundamental matrix Ψ ( t ). We make a general remark that may help in ﬁnding this general solution x ( t ). Say for the moment that A has n distinct eigenvalues r 1 , r 2 , ..., r n . Denote ξ ( k ) as an eigenvector corresponding to r k , and consider the matrices T = ξ 11 ξ 12 ··· ξ 1 n ξ 21 ξ

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lecture_39 - MA 36600 LECTURE NOTES WEDNESDAY APRIL 29...

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