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final_review (dragged) - MA 36600 FINAL REVIEW Chapter 7...

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MA 36600 FINAL REVIEW Chapter 7 § 7.1: Introduction. A system of first order equations is a collection of initial value problems dx k dt = G k ( t, x 1 , x 2 , . . . , x n ) , x k ( t 0 ) = x 0 k ; k = 1 , 2 , . . . , m. We say that such a system is a system of first order linear equations if we can write G k ( t, x 1 , x 2 , . . . , x n ) = p k 1 ( t ) x 1 + p k 2 ( t ) x 2 + · · · + p kn ( t ) x n + g k ( t ); k = 1 , 2 , . . . , m. 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 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 linear equation as a system of first order equations: In general, an n th order equation is an ordinary di ff erential equation in the form y ( n ) = G t, y, y (1) , . . . , y ( n 1) where the notation y ( k ) denotes the k th derivative; and we have the initial conditions y ( t 0 ) = y 0 , y (1) ( t 0 ) = y (1) 0 , . . . y ( n 1) ( t 0 ) = y ( n 1) 0 . If we make the substitutions x 1 = y x 2 = y (1) . . . x n = y ( n 1) = x 1 = x 2 . . . x n 1 = x n x n = G ( t, x 1 , . . . , x n ) where x 0 1 = y 0 x 0 2 = y (1) 0 . . . x 0 n = y ( n 1) 0 Assume that there are as many equations as variables i.e., that
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