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Lecture 35 - Apr 1

Lecture 35 - Apr 1 - Wednesday April 1 Lecture 35...

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Wednesday April 1 Lecture 35: Diagonalizable matrices: Applications (Refers to section 6.4) Expectations: 1. Solve a system of ordinary differential equations using diagonalization. 35.1 Introduction – Suppose we wish to solve system of differential equations of the form y ’( t ) = a 11 y ( t ) + a 12 z ( t ) z ’( t ) = a 21 y ( t ) + a 22 z ( t ) (Note: Since the constant functions y ( t ) = 0, z ( t ) = 0 are obviously a solution, for what follows, we will assume y ( t ) 0 and z ( t ) 0.) We can express this system in the form of a matrix equation: y ’( t ) = a 11 a 12 y ( t ) z ’( t ) a 21 a 22 z ( t ) If we let u = u ( t ) = ( y ( t ), z ( t )) then we will express ( y ’( t ), z ’( t )) as u ’ = u ’( t ). Thus, if A represents the coefficient matrix, we can express this system of differential equations in the more succinct form u ’( t ) = A u ( t ) Case 1 : The matrix A if a diagonal matrix . If the matrix A is diagonal this system would be easy to solve. We would have as system y ’( t ) = a 11 y ( t ) z ’( t ) = a 22 z ( t ) So obtain y ( t ) and z ( t

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