270_pdfsam_math 54 differential equation solutions odd

270_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 17. Expressed in operator notation, this system becomes ( D 2 +5)[ x ] 4[ y ]=0 , [ x ]+( D 2 +2)[ y . In order to eliminate the function x ( t ), we apply the operator ( D 2 +5) to the second equation. Thus, we have ( D 2 +5 ) [ x ] 4[ y , ( D 2 ) [ x ]+ ( D 2 )( D 2 +2 ) [ y . Adding these two equations together yields the diFerential equation involving the single func- tion y ( t )g ivenby ± ( D 2 +5)( D 2 +2) 4 ² [ y ( D 4 +7 D 2 +6 ) [ y . The auxiliary equation for this homogeneous equation, r 4 r 2 +6=( r 2 +1)( r 2 +6)=0, has roots r = ± i , ± i 6. Thus, a general solution is y ( t )= C 1 sin t + C 2 cos t + C 3 sin 6 t + C 4 cos 6 t. We must now ±nd a function x ( t ) that satis±es the system of diFerential equations given in the problem. To do this we solve the second equation of the system of diFerential equations
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