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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 ] = sin t. In order to eliminate the function y ( t ), we will apply the operator ( D 1) to the first equation above and the operator D to the second one. Thus, we have ( D 1)( D + 2)[ 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 differential equation involving the single function x ( t ) given by ( D 2 + D 2) ( D 2 D ) [ x ] = cos t 2( D 1)[ x ] = cos t. (5.3) This is a linear first order differential equation with constant coeﬃcients and so can be solved by the methods of Chapter 2. (See Section 2.3.)
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