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2 MA 36600 LECTURE NOTES: FRIDAY, APRIL 3 Equivalently, we say it is linear if we have a system in the form x ° 1 = p 11 ( t ) x 1 + p 12 ( t ) x 2 + ··· + p 1 n ( t ) x n + g 1 ( t ) x ° 2 = p 21 ( t ) x 1 + p 22 ( t ) x 2 + ··· + p 2 n ( t ) x n + g 2 ( t ) . . . . . . x ° m = p m 1 ( t ) x 1 + p m 2 ( t ) x 2 + ··· + p mn ( t ) x n + g m ( t ) Note that there are m equations and n variables. If at least one of the G k ( t, x 1 ,x 2 ,...,x n ) is not in the form above, we call the system nonlinear . We remark that we can always express an n th order linear equation as a system of Frst order linear equations. We call a linear system homogeneous if each g k ( t ) is the zero function. Otherwise, we call the system nonhomogeneous . We can always express an n th order homogeneous equation as a system of Frst order homogeneous equations. Systems of First Order Linear Equations Existence and Uniqueness Theorems. We will eventually give techniques for solving systems of Frst order equations, but for now we discuss when solutions exist. Indeed, if solutions do not exist it does not make sense to try and Fnd them!
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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