Chapter 517.Expressed in operator notation, this system becomes(D2+ 5) [x]−4[y]=0,−[x] + (D2+ 2) [y]=0.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]=0,−(D2+ 5)[x] +(D2+ 5) (D2+ 2)[y]=0.Adding these two equations together yields the differential equation involving the single func-tiony(t) given by(D2+ 5)(D2+ 2)−4[y] = 0⇒(D4+ 7D2+ 6)[y] = 0.The auxiliary equation for this homogeneous equation,r4+ 7r2+ 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 find a functionx(t) that satisfies the system of differential equations given inthe problem. To do this we solve the second equation of the system of differential equations
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