HW 12 - C . C. Let A M n ( R ) be diagonalizable with P 1...

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MATH 225 HOMEWORK 12 pp. 540 – 541 #3, 6, 11, 20 pp. 545 #3, 4, 8, 11, 12 pp. 560 – 561 #12, 16 – 18, 26, 27 pp. 571 – 572 #4; 8; 15 pp. 576 – 577 #2, 3, 7, 8 A. Find the general solution of the system Y' = AY for A = I n + E 21 + E 32 + + E n n –1 . Thus A has 1's on its diagonal, 1's just below the diagonal, and zero entries elsewhere. Writing Y = ( y 1 , . . . , y n ) T , describe the general solution by indicating how each y j ( x ) is related to y 1 ( x ). B. Consider the system Y' = AY + ( e x , e x , e x , e x , e x , e x ) T for A M 6 ( R ) and assume that det( I 6 x A ) = ( x 2 – 4)( x 2 – 9)( x 2 – 16). Show that there is a particular solution g ( x ) of this system so that g ( x ) = e ax C for some C R 6 . Determine which a R can occur and find an expression for
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Unformatted text preview: C . C. Let A M n ( R ) be diagonalizable with P 1 AP = D = diag( d 1 , . . . , d n ) Show that Y ( x ) C' ( I ) n solves the system Y' = AY + B ( x ) with B ( x ) C ( I ) n Z ( x ) = P 1 Y ( x ) solves the system Z '( x ) = D Z( x ) + P 1 B ( x ). Note that this latter system is a diagonal system that reduces finding Y to solving n first order LDEs. Use this approach to find a specific solution of the system: y 1 ' = 2 y 2 + 1/(1 + e x ) y 2 ' = y 1 + 3 y 2 + 1/(1 + e 2 x ) Hand In 12A) pp. 540-541 #6; 20; p. 545 #3; 8; p. 560-561 #12; 18 12B) p. 545 #4; p. 571572 #8; 15; A Due Friday December 2 ....
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