266_pdfsam_math 54 differential equation solutions odd

266_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 Thus, a general solution to the given system is u ( t )= c 1 1 2 c 2 e t + 1 2 e t + 5 3 t, v ( t c 1 + c 2 e t + 5 3 t. 9. Expressed in operator notation, this system becomes ( D +2)[ x ]+ D [ y ]=0 , ( D 1)[ x ]+( D 1)[ y ]=s i n t. In order to eliminate the function y ( t ), we will apply the operator ( D 1) to the Frst equation above and the operator D to the second one. Thus, we have ( D 1)( D x D 1) D [ y ]=( D 1)[0] = 0 , D ( D 1)[ x ] D ( D 1)[ y ]= D [sin t cos t. Adding these two equations yields the di±erential equation involving the single function x ( t ) given by ± ( D 2 + D 2) ( D 2 D ) ² [ x cos t 2( D 1)[ x cos t. (5.3) This is a linear Frst order di±erential equation with constant coefficients and so can be solved by the methods of Chapter 2. (See Section 2.3.) However, we will use the methods of
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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