Chapter 517.Expressed in operator notation, this system becomes(D2+5)[x]−4[y]=0,−[x]+(D2+2)[y.In order to eliminate the functionx(t), we apply the operator (D2+5) to the second equation.Thus, we have(D2+5)[x]−4[y,−(D2)[x]+(D2)(D2+2)[y.Adding these two equations together yields the diFerential equation involving the single func-tiony(t)givenby±(D2+5)(D2+2)−4²[y⇒(D4+7D2+6)[y.The auxiliary equation for this homogeneous equation,r4r2+6=(r2+1)(r2+6)=0,has rootsr=±i,±i√6. Thus, a general solution isy(t)=C1sint+C2cost+C3sin√6t+C4cos√6t.We must now ±nd a functionx(t) that satis±es the system of diFerential equations given inthe problem. To do this we solve the second equation of the system of diFerential equations
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