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One thousand + . 1(1. 3000) 1. 2300 2(. 2) + 1. 2300 = 1. 6300 the rest of the entries are got by way of carrying on with during this demeanour and we discover eventually 2/(. five) = 1. 8315. (6) The equation is linear and has integrating issue e~x. fixing, we discover y = 3ex — 2x — 2. Then whilst * = . five, y = 3e-5 — three = 3(1. 6487) — three = 1. 9461. higher accuracy should be acquired by utilizing smaller values of h or continuing as in challenge 2. forty five that is a amendment of the tactic. 2. forty five. convey find out how to enhance the accuracy of the tactic of challenge 2. forty four. the tactic which we will use is largely just like that of challenge 2. forty four other than that we receive superior values of y and y', which we denote by means of y\,y'\', y2,y21 and so on. , as indicated within the desk under. luck is accomplished whilst there's little or no distinction among the enhanced values of y [see columns headed 3/3 and 3/4 respectively within the table]. the 1st line within the desk under is equal to the 1st line within the desk of challenge 2. forty four and is located in precisely an identical method. equally the entries comparable to y1 and y[ in line of the desk under are just like these in line of the desk of challenge 2. forty four. We now refer merely to the desk given less than. As visible from this desk, the slopes reminiscent of * = . zero and x = . 1 are 1. 0000 and 1. 3000 respectively. Then the common [mean] slope is ¿(1. 0000 +1. 3000) = 1. 1500. utilizing this converted slope, we discover for the corresponding price °f y 2/2 = 1. 0000 + . 1(1. 1500) = 1. 1150 (1) The slope comparable to this price of y is given by means of y'2 = 2(. l) + 1. 1150 = 1. 3150 utilizing the given differential equation. (2) From the enhanced slope (2) we now receive a far better regular slope ¿(1. 0000 + 1. 3150) = 1. 1575 (S) and from this a better worth of y given by means of ya = 1. 0000 + (. 1)(1. 1575) = 1. 1158 (¿) which in flip offers a stronger slope equivalent to ya = 2(. l) + 1. 1158 = 1. 3158 (5) the enhanced regular slope utilizing (5) is then ¿(1. 0000 + 1. 3158) = 1. 1579 and so i/4 = 1. 0000 + (. 1)(1. 1579) = 1. 1158 (6) because this concurs with (4), the method ends. We now use yt = 1. 1158 and y'z = 1. 3158 to procure the entries within the 3rd line. and yl (line three) = 1. 1158 + (. 1)(1. 3158) = 1. 2474 y[ (line three) = 2(. 2) + 1. 2474 = 1. 6474 therefore CHAP. 2] usual DIFFERENTIAL EQUATIONS sixty three carrying on with during this demeanour we ultimately receive y(. five) = 1. 9483, the final access of the final line within the desk. This consents heavily with the genuine worth 1. 9461. * Vi y'i Vz VÍ Vi Vs 2/4 . zero . 1 . 2 . three . four . five 1. 0000 1. one thousand 1. 2474 1. 4313 1. 6557 1. 9247 1. 0000 1. 3000 1. 6474 2. 0313 2. 4557 2. 9247 1. 1150 1. 2640 1. 4496 1. 6759 1. 9471 1. 3150 1. 6640 2. 0496 2. 4759 2. 9471 1. 1158 1. 2648 1. 4505 1. 6769 1. 9482 1. 3158 1. 6648 2. 0505 2. 4769 2. 9482 1. 1158 1. 2648 1. 4506 1. 6770 1. 9483 2. forty six. remedy challenge 2. forty four by way of the Taylor sequence procedure. via successively differentiating the given differential equation, now we have y' = Zx + y, y" = 2 + y', y'" = y", yW = y'", y<^ = j/«v>, hence y(0) = 1, y'(0) = 1, y" = three, y'" = . three, i/«v> = three, y = three, and so Then whilst x = . five, 3/(. five) 2.

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