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Section 9.1 & 9.3

# Section 9.1 & 9.3 - Differential Equations Math 308...

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Differential Equations Texas A&M University Math 308, Spring 2008 c circlecopyrt F. Dos Reis Sections 9.1, 9.3 Exercise 1. Express the given systems of differential equation in matrix notation. braceleftbigg x ( t ) = 7 x + 2 y y ( t ) = 3 x - 2 y x 1 ( t ) = x 1 - x 2 + x 3 - x 4 x 2 ( t ) = x 1 + x 4 x 3 ( t ) = tx 1 + sin( t ) x 3 x 4 ( t ) = 0 Exercise 2. Express the given differential equations as matrix systems in normal form. (1 - t 2 ) y ′′ ( t ) - 2 ty ( t ) + 2 y ( t ) = 0 y ′′′ + 2 ty ′′ - 3 y + t 2 y = 0 Recall: Determinant: Let A be an n × n matrix. Te following state- ments are equivalent: A is inversible the determinant of A is not 0. the only solution to AX = 0 is X = 0. The columnum (row) form a linearly independant set. Exercise 3. Compute the determinants of A ( t ) = bracketleftbigg 1 e 2 t t e 3 t bracketrightbigg and, of B ( t ) = e 3 t 1 t 3 e 3 t 0 1 9 e 3 t 0 0 Definition: Let A ( t ) be a matrix such that the entries are functions of t . A ( t ) is continuous if all the entries are continuous. A ( t ) is differentiable if all the entries ( a i,j ( t )) are differentiable. In this case A ( t ) is the matrix with entries

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Section 9.1 & 9.3 - Differential Equations Math 308...

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