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

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MA 36600 LECTURE NOTES: FRIDAY, APRIL 3 Systems of First Order Linear Equations Recap. Recall that a frst order equation is an ordinary diferential equation in the Form y ° = G ( t, y ) ,y ( t 0 )= y 0 . In general, an n th order equation is an ordinary diferential 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 diferential 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±erential Equations. We can always express an n th order equation as a system oF ±rst order equations. We explain why. Make the Following substitutions: x
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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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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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