Example A The second-order differential equation dy dy 3 + 2y = 0 dx (4.114) dæ? has the general solution y(x) c1e* + c2e2", (4.115) where c1 and c2 are arbitrary constants. The difference equation associated with this differential equation is Yk+2 – 3yk+1 +2yk = 0. (4.116) Its general solution is Yk = A+ B2k, (4.117) LINEAR DIFFERENCE EQUATIONS 131 since the characteristic equation r2 – 3r+2 = 0 has roots ri = 1 and r2 = 2; A and B are arbitrary constants. We now show how the result given by equation (4.117) can be obtained from equation (4.115). Let us calculate Dky(x); it is D*y(x) dn (c1e + cze2") = cie" +c22*e2. (4.118) dæn Therefore, D*y(x)\a=0 : C1 + c22k. (4.119) Yk which is the same as equation (4.117) except for the labeling of the arbitrary constants.

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