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lecture_32

# lecture_32 - MA 36600 LECTURE NOTES MONDAY APRIL 13 Basic...

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MA 36600 LECTURE NOTES: MONDAY, APRIL 13 Basic Theory of Systems of First Order Linear Equations Solutions to Systems of Linear Differential Equations. We continue to focus on a system of first order differential equations in the form x 0 1 = p 11 ( t ) x 1 + p 12 ( t ) x 2 + · · · + p 1 n ( t ) x n + g 1 ( t ) x 0 2 = p 21 ( t ) x 1 + p 22 ( t ) x 2 + · · · + p 2 n ( t ) x n + g 2 ( t ) . . . . . . x 0 m = p m 1 ( t ) x 1 + p m 2 ( t ) x 2 + · · · + p mn ( t ) x n + g m ( t ) Such a system contains m equations and n variables. When m = n , we can express this as d dt x = P ( t ) x + g ( t ) . We focus on a few special cases to gain intuition about the solutions of this system. Example #1. Consider first when m = n = 1. The first order equation x 0 = p ( t ) x + g ( t ) can be solved using integrating factors: μ ( t ) = exp - Z t p ( τ ) = x ( t ) = 1 μ ( t ) Z t μ ( τ ) g ( τ ) + C . Hence the general solution is in the form x ( t ) = C x (1) ( t ) + x ( p ) ( t ), where x (1) ( t ) = 1 μ ( t ) = exp Z t p ( τ ) = d dt x (1) = p ( t ) x (1) is a solution to the homogeneous equation, and x ( p ) ( t ) = 1 μ ( t ) Z t μ ( τ ) g ( τ ) is a particular solution to the nonhomogeneous equation. Example #2. Consider now when m = n = 2. The equation x 00 = q ( t ) x + p ( t ) x 0 + g ( t ) can be transformed into a system of first order equations through the substitution x = x x 0 = d dt x = 0 1 q ( t ) p ( t ) x + 0 g ( t ) . Say that we have three functions x 1 = x 1 ( t ), x 2 = x 2 ( t ), and X = X ( t ) such that i. x 1 and x 2 are solutions to the homogeneous equation x 00 - p ( t ) x 0 - q ( t ) x = 0 . ii. Their Wronskian is nonzero, in terms of the function W ( x 1 , x 2 ) ( t ) = x 1 ( t ) x 0 2 ( t ) - x 0 1 ( t ) x 2 ( t ) = det x 1 ( t ) x 2 ( t ) x 0 1 ( t ) x 0 2 ( t ) = C exp Z t p ( τ ) .

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