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# 285lect16 - 1 Lecture 16 1.1 Linear ODE of order n In a...

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Unformatted text preview: 1 Lecture 16 1.1 Linear ODE of order n In a similar way, we can consider a linear ODE of order n : y ( n ) + p 1 ( x ) y ( n &amp;amp; 1) + p 2 ( x ) y ( n &amp;amp; 2) + ::: + p n ( x ) y ( x ) = f ( x ) with the initial conditions: y ( x ) = b ; y ( x ) = b 1 ; :::; y ( n &amp;amp; 1) ( x ) = b n &amp;amp; 1 . We have the following Theorem 1 (Existence and uniqueness of solutions) Consider the ini- tial value problem y ( n ) + p 1 ( x ) y ( n &amp;amp; 1) + p 2 ( x ) y ( n &amp;amp; 2) + ::: + p n ( x ) y ( x ) = f ( x ) ; y ( x ) = b ; y ( x ) = b 1 ; (1) :::; (2) y ( n &amp;amp; 1) ( x ) = b n &amp;amp; 1 . (3) Suppose that the functions p j ( x ) ; j = 1 ; 2 ; :::; n; and f ( x ) are continuous on an open interval I = ( x &amp;amp; &amp;quot; 1 ; x + &amp;quot; 2 ) containing x . Then there is a unique solution y ( x ) ; x &amp;amp; &amp;quot; 1 &amp;lt; x &amp;lt; x + &amp;quot; 2 , of the initial value problem (3). De&amp;amp;nition 2 Functions f 1 ( x ) ; f 2 ( x ) ; :::; f n ( x ) ; a &amp;lt; x &amp;lt; b; are said to be linearly dependent if there are constants &amp;amp; 1 ; &amp;amp; 2 ; :::; &amp;amp; n such that &amp;amp; 2 1 + &amp;amp; 2 2 + ::: + &amp;amp; 2 n 6 = 0 and &amp;amp; 1 f 1 ( x ) + &amp;amp; 2 f 2 ( x ) + ::: + &amp;amp; n f n ( x ) , for every a &amp;lt; x &amp;lt; b . Otherwise functions f 1 ( x ) ; f 2 ( x ) ; :::; f n ( x ) ; a &amp;lt; x &amp;lt; b; are called linearly independent....
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285lect16 - 1 Lecture 16 1.1 Linear ODE of order n In a...

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