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# lecture_32 (dragged) 1 - 2 MA 36600 LECTURE NOTES MONDAY...

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2 MA 36600 LECTURE NOTES: MONDAY, APRIL 13 Then the general solution to the nonhomogeneous equation is in the form x ( t )= c 1 x 1 ( t )+ c 2 x 2 ( t )+ X ( t ) where c 1 and c 2 are constants. In terms of matrices, we see that the general solution is in the form x ( t )= c 1 x (1) ( t )+ c 2 x (2) ( t )+ x ( p ) ( t ) , where we have denoted the vectors x (1) ( t )= ° x 1 ( t ) x ° 1 ( t ) ± , x (2) ( t )= ° x 2 ( t ) x ° 2 ( t ) ± , x ( p ) ( t )= ° X ( t ) X ° ( t ) ± . We can express the general solution to the homogeneous equation in the form x ( c ) ( t )= c 1 x (1) ( t )+ c 2 x (2) ( t )= Ψ ( t ) c where Ψ ( t )= ° x 1 ( t ) x 2 ( t ) x ° 1 ( t ) x ° 2 ( t ) ± , c = ° c 1 c 2 ± . Note that W ( t )=det Ψ ( t ) is the Wronskian. Fundamental Set of Solutions. Consider the general system of Frst order linear equations d dt x = P ( t ) x + g ( t ) . Say that we can Fnd ( n + 1) matrix functions x (1) ( t ) ,..., x ( n ) ( t ) , x ( p ) ( t ) such that i. The x ( k ) are solutions to the homogeneous equation d dt x ( k ) = P ( t ) x ( k ) ,k =1 , 2 ,...,n . ii. The n × n matrix Ψ ( t )= x 11 ( t ) x 12 ( t ) ··· x 1 n ( t ) x 21 ( t ) x 22 ( t ) ··· x 2 n ( t ) .
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