Chapter 6Equation (6.5) will equal zero and, therefore, the differential equation will be satisfied forr=±1 andr= 2. Thus, three solutions to the differential equation arey=x,y=x−1,andy=x2.Since these functions are linearly independent, they form a fundamentalsolution set.(b)Lety(x) =xr. In addition to (6.4), we need the fourth derivative ofy(x).y(4)= (y) =r(r−1)(r−2)(r−3)xr−4= (r4−6r3+ 11r2−6r)xr−4.Thus, ify=xris a solution to this fourth order Cauchy-Euler equation, then we musthavex4(r4−6r3+ 11r2−6r)xr−4+ 6x3(r3−3r2+ 2r)xr−3+2x2(r2−r)xr−2−4xrxr−1+ 4xr= 0⇒(r4−6r3+ 11r2−6r)xr+ 6(r3−3r2+ 2r)xr+ 2(r2−r)xr−4rxr+ 4xr= 0⇒(r4−5r2+ 4)xr= 0.(6.6)Therefore, in order fory=xrto be a solution to the equation withx >0, we must haver4−5r2+ 4 = 0. Factoring this equation yieldsr4−5r2+ 4 = (r2−4)(r2−1) = (r−2)(r+ 2)(r−1)(r+ 1) = 0.Equation (6.6) will be satisfied if
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