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MIT18_03S10_ls5

# MIT18_03S10_ls5 - LS.5 Theory of Linear Systems 1 General...

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± LS.5 Theory of Linear Systems 1. General linear ODE systems and independent solutions. We have studied the homogeneous system of ODE’s with constant coeﬃcients, (1) x = A x , where A is an n × n matrix of constants ( n = 2 , 3). We described how to calculate the eigenvalues and corresponding eigenvectors for the matrix A , and how to use them to find n independent solutions to the system (1). With this concrete experience solving low-order systems with constant coeﬃcients, what can be said in general when the coeﬃcients are not constant, but functions of the independent variable t ? We can still write the linear system in the matrix form (1), but now the matrix entries will be functions of t : x = a ( t ) x + b ( t ) y ± ± a ( t ) b ( t ) � ± x x (2) y = c ( t ) x + d ( t ) y , y = c ( t ) d ( t ) y , or in more abridged notation, valid for n × n linear homogeneous systems, (3) x = A ( t ) x . Note how the matrix becomes a function of t we call it a “matrix-valued func- tion” of t , since to each value of t the function rule assigns a matrix: a ( t 0 ) b ( t 0 ) t 0 A ( t 0 ) = c ( t 0 ) d ( t 0 ) In the rest of this chapter we will often not write the variable t explicitly, but it is always understood that the matrix entries are functions of t . We will sometimes use n = 2 or 3 in the statements and examples in order to simplify the exposition, but the definitions, results, and the arguments which prove them are essentially the same for higher values of n . Definition 5.1 Solutions x 1 ( t ) , . . . , x n ( t ) to (3) are called linearly dependent if there are constants c i , not all of which are 0, such that (4) c 1 x 1 ( t ) + . . . + c n x n ( t ) = 0 , for all t. If there is no such relation, i.e., if (5) c 1 x 1 ( t ) + . . . + c n x n ( t ) = 0 for all t all c i = 0 , the solutions are called linearly independent , or simply independent . The phrase “for all t is often in practice omitted, as being understood. This can lead to ambiguity; to avoid it, we will use the symbol 0 for identically 0 , meaning: “zero for all t ”; the symbol →� 0 means “not identically 0”, i.e., there is some t -value for which it is not zero. For example, (4) would be written 21

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22 18.03 NOTES: LS. LINEAR SYSTEMS c 1 x 1 ( t ) + . . . + c n x n ( t ) 0 .
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MIT18_03S10_ls5 - LS.5 Theory of Linear Systems 1 General...

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