testreview3

testreview3 - 5. Given the differential equation 2 2 3 (D...

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Math 2090, Fall 2009. Testreview3. Review sections: 5.1, 5.3, 5.6, 5.7, 6.1-6.3, and 6.7. 1. Let 2 : ( ) T M R R be the transformation defined by ( ) det( ) T A A = . Show that T is a linear transformation or show, by example, which property for linear transformations fails. 2. Let 4 3 : T R R be the linear transformation defined by 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , ) ( 3 2 ,3 10 4 6 ,2 5 6 ) T x x x x x x x x x x x x x x x x = + - + + - + + - - (a) Find the matrix, A , for T; (b) Find a basis for the ( ) Ker T and its dimension; 3. Let 6 3 4 A 5 2 2 0 0 1 - = - - - .

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(a) Find the eigenvalues for A . (b) For each eigenvalue, find a basis for the eigenspace corresponding to the eigenvalue, and state its dimension. (c) Is A a defective or non defective matrix? Explain your answer. 4. Determine three LI solutions to the DE 3 2 2 2 0, 0 x y x y xy y x ′′′ ′′ + - + = of the form ( 29 r x x y = , and thereby determine the general solution.

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Unformatted text preview: 5. Given the differential equation 2 2 3 (D 4)(D 4D 13) y--+ = , (a) Find a basis for the solution space; (b) Find the general solution to the differential equation. 6. Given the differential equation ( 29 3 1 5 x D y e-+ = (a) Find the complementary solution, c y , to the corresponding homogeneous differential equation; (b) Find the general solution to the given non-homogeneous equation 7. Solve the following IVP ( 29 ( 29 2 10sin 2, 1 y y y t y y ′′ ′ +-= - ′ = = 8. Use the method of variation of parameters to find a particular solution to the given differential equation (a) 2 2 4 4 , x y y y x e x--′′ ′ + + = (b) 2 2 , 1 1 x e y y y x x-′′ ′ + + = <-...
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This note was uploaded on 04/21/2010 for the course MATH 2090 taught by Professor Staff during the Fall '08 term at LSU.

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testreview3 - 5. Given the differential equation 2 2 3 (D...

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