Advanced Mathematics by C. B. Gupta

By C. B. Gupta

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This formula is called the Stirling’s difference formula. yu = y0 + u SOLVED EXAMPLES Example 1. Given u0 = 580, u1 = 556, u2 = 520, u3 = —, u4 = 384, find u3. Solution. Let the missing term u3 = X 26 ADVANCED MATHEMATICS ∴ The forward difference table is x ux 0 580 1 556 ∆u x ∆2u x ∆ 3u x ∆ 4 ux – 24 – 12 – 36 2 X – 472 520 X – 484 1860 – 4X X – 520 3 1388 – X X 904 – 2X 384 – X 4 384 Here four values of ux are given. Therefore, we can assume ux to be a polynomial of degree 3 in x ∴ ∆ 4 ux = 0 or or 1860 – 4X = 0 X = 465.

X 0 1 4 6 yx 1 –1 1 –1 36 ADVANCED MATHEMATICS Solution. Here and x0 = 0, x1 = 1, x2 = 4, x3 = 6 y0 = 1, y1 = – 1, y2 = 1, y3 = – 1 Putting the above values in Lagrange’s formula, we get f (x) = ( x − 0) ( x − 1) ( x − 6) ( x − 1) ( x − 4) ( x − 6) ( x − 0) ( x − 4) ( x − 6) 1+ (–1) + 1 ( 4 − 0) (4 − 1) (4 − 6) ( 0 − 1) (0 − 4) (0 − 6) (1 − 0) (1 − 4) (1 − 6) + =– ( x − 0) ( x − 1) ( x − 4) (–1) (6 − 0) (6 − 1) (6 − 4) 1 3 1 1 3 [x3 – 11x2 + 34x – 24] – [x – 10x2 + 24x] – [x – 7x2 + 6x] 15 24 24 – =– 1 [x3 – 5x2 + 4x] 60 1 3 3 2 10 x + x – x + 1.

Given u0 = 3, u1 = 12, u2 = 81, u3 = 200, u4 = 100, u5 = 8 find ∆5 u0 without forming difference table. 15. If f(0) = – 3, f(1) = 6, f(2) = 8, f (3) = 12 and the third difference being constant, find f(6). 16. Represent the function f(x) = 2x3 – 3x2 + 3x – 10 and its successive differences into factorial notation. 17. Find the function whose first difference is x3 + 3x2 + 5x + 12. 18. Obtain the function whose first difference is: (i) ex (iv) x(2) + 5x (ii) x(x – 1) (iii) a (v) sin x (vi) 5x. 20 ADVANCED MATHEMATICS 19.

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