By Gupta, C.B.,Malik, A.K. , Kumar, Vipin
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Extra resources for Advanced Mathematics As per IV Semester of RTU & Other Universities
En – nC1 En –1 + nC2 En–2 – ..... S. 1 1. Prove that ∆2 ≡ E2 – 2E + 1. 2. Prove that if f(x) and g(x) are the function of x then (i) ∆[f(x) + g(x)] = ∆f(x) + ∆g(x) (ii) ∆[af(x)] = a ∆f(x) (iii) ∆[f(x) g(x)] = f(x) ∆ g(x) + g(x + 1) f(x) = f(x + 1) ∆ g(x) + g(x) ∆f(x) (iv) ∆ LM f ( x) OP = g( x) ∆ f ( x) − f ( x) ∆ g( x) . g( x ) g( x + h) N g( x) Q 3. Evaluate (i) ∆[sinh (a + bx)] (iii) ∆ [cot 2x ] (ii) ∆[tan ax] (iv) ∆ (x + cos x) 19 CALCULUS OF FINITE DIFFERENCES (v) ∆ (x2 + ex + 2) (vi) ∆ [log x] LM x OP N cos 2 x Q 2 (vii) ∆ [eax log bx] (viii) ∆ 4.
Example 22. Prove that u0 + nC1 u1x + nC2 u2x2 + ..... + unxn = (1 + x)n u0 + nC1 (1 + x)n–1 x ∆u0 + nC2 (1 + x)n–2 x2 ∆2 u0 + ..... + xn ∆n u0 Solution. S. (1 + x)n u0 + nC1 (1 + x)n–1 x∆u0 + nC2 (1 + x)n–2 x2∆2 u0 + ...... + xn∆n u0 = ((1 + x) + x∆)n u0 = (1 + x ( 1 + ∆))n u0 = (1 + xE)n u0 = (1 + nC1 xE + nC2 x2 E2 + nC3 x3 E3 + ..... + xn E n ) u0 = u0 + nC1 u1 x + nC2 u2 x2 + nC3 u3 x3 + ..... S. Example 23. + (– 1)n ux. S. ux +n – nC1 ux + n – 1 + nC2 ux + n–2 – ..... + (– 1)n ux Solution.
2. GAUSS ELIMINATION METHOD In this method, the unknowns from the system of equations are eliminated successively such that system of equations is reduced to an upper triangular system from which the unknowns are determined by back substitution. We proceed stepwise as follows. Consider the system of equations a11 x1 + a12 x 2 + ...... + a1n x n = b1 a21 x1 + a22 x 2 + ...... + a2 n x n = b2 . . : : : : am1 x1 + am 2 x 2 + ...... 1) Step 1. To eliminate x1 from the second, third .... nth equation a21 times the a11 first equation from the second equation, similarly we eliminate x1 from third equation by subtracting a31 times the first equation from the third equation, etc.
Advanced Mathematics As per IV Semester of RTU & Other Universities by Gupta, C.B.,Malik, A.K. , Kumar, Vipin