C H A P T E R 2 0D I F F E R E N C E E Q U A T I O N S6120.52.(a) The characteristic equationm2+2m+1=(m+1)2=0 has the double rootm= −1, so thegeneral solution of the homogeneous equation isxt=(C1+C2t)(−1)t. We find a particular solution byinsertingu∗t=A2t. This yieldsA=1, and so the general solution of the inhomogeneous equation isxt=(C1+C2t)(−1)t+2t.(b) By using the method of undetermined coefficients to determine the constantsA,B, andCin theparticular solutionu∗t=A5t+Bcosπ2t+Csinπ2t, we obtainA=14,B=310, andC=110. So thegeneral solution to the given equation isxt=C1+C22t+145t+310cosπ2t+110sinπ2t.4.Thecharacteristicequationism2−4(ab+1)m+4a2b2=0, withsolutionsm1,2=2(ab+1±√1+2ab ).The general solution is thereforeDn=C1mn1+C2mn2.6.Insertingxt=ut(−a/2)tinto [20.21],assuming thatb=14a2,we obtain the equationxt+2+axt+1+14a2xt=ut+2(−a/2)t+2+aut+1(−a/2)t+1+14a2ut(−a/2)t=14a2(−a/2)t(ut+2−2ut+1+ut), which is 0 ifut+2−2ut+1+ut=0. The general solution of thisequation isut=A+B t, soxt=ut(−a/2)t=(A+B t)(−a/2)t, which is the result claimed for case2 in the frame on page 751.8.(a) The first two equations state that consumption and capital are proportional to the net national productin the previous period. The third equation states that net national product,Yt, is divided between con-sumption,Ct, and net investment,Kt−Kt−1.