notes 4-1 - SECTION 4.1 Nth ORDER LINEAR STUFF THE...

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SECTION 4.1 Nth ORDER LINEAR STUFF THE OPERATOR. L [ y ] = y ( n ) + p 1 ( t ) y ( n - 1) + p 2 ( t ) y ( n - 2) + · · · + p n ( t ) y , or L [ y ] = ( D n + p 1 ( t ) D n - 1 + p 2 ( t ) D n - 2 + · · · + p n - 1 ( t ) D + p n ( t ))[ y ] . PROPERTIES OF THE OPERATOR. First, as for order two, the operator L is linear, that is, L [ c 1 y 1 + c 2 y 2 + · · · c n y n ] = c 1 L [ y 1 ] + c 2 L [ y 2 ] = · · · c n L [ y n ] . Second, operators with constant coefficients behave just like polynomials. To see what this means, let’s compute (3 D + 2)[(2 D - 1)[ y ]]. INITIAL CONDITIONS. y ( t 0 ) = y 0 , y 0 ( t 0 ) = y 1 , . . . , y ( n - 1) ( t 0 ) = y n - 1 EXISTENCE AND UNIQUENESS. When p 1 ( t ) ,p 2 ( t ) ,...p n ( t ) and g ( t ) are all con- tinuous on an interval I and t 0 is any point in I, then there is a unique function y ( t ) that satisfies the initial conditions above and the differential equation L [ y ] = g ( t ) on I, and this function y ( t ) is defined throughout the interval I. GENERAL SOLUTIONS, HOMOGENEOUS CASE.
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This note was uploaded on 05/19/2010 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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notes 4-1 - SECTION 4.1 Nth ORDER LINEAR STUFF THE...

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