Download e-book for kindle: A text book of engineering mathematics Volume 2 by Rajesh Pandey

By Rajesh Pandey

ISBN-10: 1441657983

ISBN-13: 9781441657985

ISBN-10: 9380257120

ISBN-13: 9789380257129

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1 eX (D _1)2 (D2 + 1)2 1 1 eX (D - 1)2 (12 + 1)2 1 1 X -e (D _1)2 (2)2 1 1 X -e (D _1)2 4 = eX 1 1 (D + 1_1)2 4 43 A Textbook ofEn~neerin~ Mathematics Volume - II = eX ~ ! D2 4 = ! eX ~ (1) = 4 D2 2 eX x 4 2 ! = .!. x2 eX 8 :', The required solution is y = C. 1. y = (Cl or X + C2) ex + (C3 x + ~) cos x + (Cs x + C6) sinx + .!. 8 x2 eX Example 3. Solve (D + 2) (D -1)3 Y = eX Solution. Here the auxiliary equation is (m + 2) (m - 1)3 = 0 m = - 2 and m = 1 (thrice) or Therefore C. 1. = = 1 eX (D + 2) (D _1)3 1 eX (1 + 2) (D - 1)3 1 1eX1 __ = _ eX 1 1 3 (D _1)3 3 {(D + 1) -I} ~ :.

Example 6. Solve (2x + y - 3) dy = (x + 2y - 3) dx dy Solution. The given differential equation is dx 14 = x + 2y-3 ---=-2x + Y - 3 (1) Di(ferential Equations o(First Order and First Degree pu tting x = X + hand y = Y + k (2) dY (X + h) + 2 (Y + k) -3 the equation (1) reduce to - = -"----'----------'-dX 2 (X + h) + (Y + k) -3 dY = X + 2Y + (h + 2K - 3) dX 2X + Y + (2h + K - 3) or (3) Choose hand k such that h + 2k - 3 = 0 and 2h + k - 3 = 0 (4) Solving the equations (4) we have h = 1 = k :. From (2) x = X + 1 and y = Y + 1 or X = x - 1 and Y = Y - 1 (5) dY X + 2Y Also (3) reduce to - = - - dX 2X+ Y .

I. 1. P. P. P. P. P. P. of %(-;) (i cos ax - sin ax) 1 x =- - - cos ax 2 a 1. --:---:- sm ax = .. I. P. of = - x 2a 2 1 D +a . 2 e,ax -~ (~) (i cos ax - sin ax) . sIn ax 1 x . ---::---:-- cos ax = - sm ax .. D2 + a 2 2a Example 4. Solve (D2 + D + 1) Y = sin 2x Solution. Here the auxiliary equation is m 2 + m + 1 = 0 which gives m :. P. 1. = 2 21 sin 2x replacing D2 by - 22 (-2) +D+1 = -1- D-3 . 2x sm 1 (D - 3) (D + 3) 21 D -9 (D + 3) sin 2x (D + 3) sin 2x = ~ 21 -9 (D + 3) sin 2x = -~ (D + 3) sin 2x = -1 [D (sin 2x) + 3 sin 2x] 13 13 = -~ [2 cos 2x + 3 sin 2x] Since D means differentiation with respect to x 13 :.

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A text book of engineering mathematics Volume 2 by Rajesh Pandey


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