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notes 4-1

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

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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. Solutions y 1 ( t ) , y 2 ( t ) , . . . y n ( t ) of L [ y ] = 0 form a fundamental set of solutions on an interval I if every solution on the interval I can be written as c 1 y 1 + c 2 y 2 + . . . + c n y n for some choice of constants c 1 , c 2 , . . . , c n . Note that a fundamental set of solutions has as many functions as the order of the differential equation. If the Wronskian of n solutions is not zero somewhere in the interval I, then the

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

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