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lecture_28 (dragged)

# lecture_28 (dragged) - x 1 = y x 2 = y(1 x n = y n − 1...

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MA 36600 LECTURE NOTES: FRIDAY, APRIL 3 Systems of First Order Linear Equations Recap. Recall that a first order equation is an ordinary di ff erential equation in the form y = G ( t, y ) , y ( t 0 ) = y 0 . 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 . Recall that this equation is a linear equation if we can write G ( t, x 1 , x 2 , . . . , x n ) = p 1 ( t ) x 1 + p 2 ( t ) x 2 + · · · + p n ( t ) x n + g ( t ) . Equivalently, we say it is linear if we can write the di ff erential equation in the form y ( n ) p n ( t ) y ( n 1) · · · p 2 ( t ) y p 1 ( t ) y = g ( t ) . Any n th order equation which cannot be placed in this form is called a nonlinear equation. Recall that this equation is homogeneous equation if g ( t ) is the zero function i.e., the right-hand side is identically zero. Otherwise, we call this a nonhomogeneous equation . Systems of First Order Di ff erential Equations. We can always express an n th order equation as a system of first order equations. We explain why. Make the following substitutions:
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Unformatted text preview: x 1 = y x 2 = y (1) . . . x n = y ( n − 1) and x 1 = y x 2 = y (1) . . . x n = y ( n − 1) That is, denote the Functions x k = y ( k − 1) and constants x k = y ( k − 1) For k = 1 , 2 , . . . , n . Then we have the Following system oF diferential equations x ° 1 = x 2 . . . x ° n − 1 = x n x ° n = G ( t, x 1 , . . . , x n ) with initial conditions x k ( t ) = x k . In general, iF we have Functions G k such that we have the system x ° k = G k ( t, x 1 , x 2 , . . . , x n ) , x k ( t ) = x k ; k = 1 , 2 , . . . , m ; we call this a system oF frst order equations . Note that there are m equations and n variables. We say that such a system is a system oF frst 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. 1...
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