By Carl M. Bender

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Nonlinear difference equations are usually harder to solve than nonlinear differential equations. However, here are some examples that can be solved in closed form. Example I) ad' = a;. To solve this equation we take the logarithm of both sides: In a,+1 2 In a•. The solution to this equation is easy to find because it IS a linear equation for In a,. The solution is In ", = 2'-' In ",. Thus, a, = If,'-·". +, Example 7 an a;"/a,, This equation can also be solved by taking logarithms of both sides: In a..

2a; - L This equation is soluble because the transcendental functions cos x and cosh x satisfy the functional relation I(2x) 2[f(xJ]2 L If Ia, I $ I, we substitute a, cos e, and if la,1 2. I We substitute a, = cosh e,. In either case, the resulting equation for e, is linear: II,. 1 20•. 3 HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS In this section we survey the methods that are commonly used to solve higherorder homogeneous linear difference equations. Our discussion will emphasize the strong parallel between differential and difference equations.

5) to be equivalent to the seemingly weaker inhomogeneous constraints y(x) bounded as x - ± 00. 4a) bounded ness of y(x) as Ixi- 00 implies that y_O as lxi-CD Example 4 Eigenvalue problem having transcendental eigenvalues. Consider the eigenvalue problem y" + (E - x)y 0 (0 < x < co), y(O)=O, y(oo) = O. 6) It will be shown in Example 5 of Sec. 6) that vanishes as x ..... 8)]. The boundary condition y(O) = 0 gives the eigenvalue condition Ai (- E) = O. The Airy function Ai (- E) is a transcendental function whose zeros may be computed numerically.