8.7.2 Example B For this example, the mixed differential-difference equation is 2 Ay(x) = x- d (Ay(x)) + Ay( (8.277) dx If Ay is replaced by z, then we obtain two coupled equations CAy= (8.278) 2 dz...

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Extracted text: 8.7.2 Example B For this example, the mixed differential-difference equation is 2 Ay(x) = x- d (Ay(x)) + Ay( (8.277) dx If Ay is replaced by z, then we obtain two coupled equations CAy= (8.278) 2 dz (--) dz Z = x- dx (8.279) dx Inspection of equation (8.279) shows it to be a Clairault-type differential equa- tion whose solution is z = cx + c², arbitrary constant. (8.280) C= Therefore, Ay = = cx + c², and Cy(æ) = cA¬!x + c²A-1 1 G) x(x – 1) + A(x)+ c²x, (² – ) x + A(x), (8.281) where A(x) is an arbitrary period-1 function of x.