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AMath350.F12.A6.Sol

# 1 3 2 and y1x 3c1 e 3x 1 c2 echoice which satises both

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Unformatted text preview: atisfy equation ( ) for all x IR we need 1 3 3 1 4 +x 1 . 1 = 3 2 and y1(x) =3c1 e. 3x 1 + c2 echoice which satisﬁes both these constraints is =) = c 12 But the only e 3x 4 0 +x 1 1 y (x e + c2 . 2 1 independent 0 IR. 1 on 1 = 2 = 0. Thus y1 , y2 are linearly Apply IC Apply IC 3 3 c e 3 + 5 c e 13 5 y(1) = 3 ) c1 e 3 + 4 c2 e 3 3 = 3 ) c1 = 2e3 , c2 = 4e3 . 2 ) c1 e 3 + 4 2 e 3 c2 1 y(1) = = 2 ) c1 = 2 e 3 , c2 = 4e 3 . 2 2 c1 e 3 + c2 e Solution of IVP Solution of IVP 1 1 1 1 + 4x 1 4 y(x) = 2e 3((x 1)) 1 + 4e 3((x 1)) +x 1 = e 3((x 1)) 1 + 4x . 3x 1 3x 1 4 1 + 4e 2 + 4x . 0 +x 1 y ( x) = 2 e =e 3x 1 1 1 2 + 4x 0 50 . Characteristic equation: ( 5 )2 = 0 ) = 5, 5. 0 5 0 . Characteristic equation: ( 5 5 )2 = 0 ) = 5, 5. 0 5 5. See attached Maple printout. The eigenvalues can 1be identiﬁed by inspection. 50 1 5 Clearly ) v1 = 1 is an eigenvector of 5. 1= 05 0 5 0= 05 Clearly ) v1 = 0 is an eigenvector of 5. 0 5 0 0 0 50 0 0 0 Also ) v2 0 is an eigenvector of 5. 0= 0 05 0 5 1= 5 Also ) v2 1 is an eigenvector of 5. 0 are linearly independent since 5 1 5 1 v1 and v2 v1 an...
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