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Unformatted text preview: Linear Systems We will now generalize some of the results from second order equations by considering (1) higher order differential equations (2) systems of first order differential equations. It will turn out that the first case can be reduced to the second, so we will focus on ways to solve systems of first order equations. In particular, we will look at linear systems and find ways to apply our knowledge of linear algebra. Systems of Equations We can express a higher order differential equation as a system of first order equations as follows. Suppose we are given a higher order (linear) equation f n ( x ) y ( n ) + f n- 1 ( x ) y ( n- 1) + + f 1 ( x ) y + f ( x ) y = 0 . To rewrite this as a system of n 1st-order equations we will (1) First we define new functions: y 1 ( x ) = y ( x ) y 2 ( x ) = y ( x ) y 3 ( x ) = y 00 ( x ) . . . y n ( x ) = y ( n- 1) ( x ) . (2) This immediately gives us our first n- 1 1st-order equations: y 1 = y 2 y 2 = y 3 ....
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