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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, APRIL 3 Systems of First Order Linear Equations Recap. Recall that a first order equation is an ordinary differential equation in the form y = G ( t,y ) , y ( t ) = y . In general, an n th order equation is an ordinary differential 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 ) = y , y (1) ( t ) = y (1) , ... y ( n- 1) ( t ) = y ( n- 1) . 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 differential 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 Differential Equations. We can always express an n th order equation as a system of first order equations. We explain why. Make the following substitutions: 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 differential equations x 1 = x 2 ....
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