Explanation Method used: Euler’s method: “Consider the differential equation d y d x = F ( x , y ) subject to the initial condition y ( x 0 ) = y 0 . To find an approximation of y ( b ) , where b is a number greater than x 0 and n is a positive integer, compute h = b − x 0 n , x 1 = x 0 + h , x 2 = x 0 + 2 h , x 3 = x 0 + 3 h ... x n = x 0 + n h = b and y 0 = y ( x 0 ) , y 1 = y 0 + h F ( x 0 , y 0 ) , ... , y n = y n − 1 + h F ( x n − 1 , y n − 1 ) . Then y n gives an approximation of the true value y ( b ) of the solution to the initial-value problem at x = b ”. Calculation: From the given information, it is observed that x 0 = 0 , n = 5 and b = 0.5 . Substitute x 0 = 0 , n = 5 and b = 0.5 in h = b − x 0 n . h = 0.5 − 0 5 = 0.1 That is, the value of h is 0.1. Compute the value of x 1 , x 2 , x 3 , x 4 and x 5 as follows. Substitute x 0 = 0 , and h = 0.1 in x 1 = x 0 + h . x 1 = 0 + 0.1 = 0.1 Substitute x 0 = 0 , and h = 0.1 in x 2 = x 0 + 2 h . x 2 = 0 + 2 ( 0.1 ) = 0.2 Substitute x 0 = 0 , and h = 0.1 in x 3 = x 0 + 3 h . x 3 = 0 + 3 ( 0.1 ) = 0.3 Substitute x 0 = 0 , and h = 0.1 in x 4 = x 0 + 4 h . x 4 = 0 + 4 ( 0.1 ) = 0.4 Substitute x 0 = 0 , and h = 0.1 in x 5 = x 0 + 5 h . x 5 = 0 + 5 ( 0.1 ) = 0.5 The given function is F ( x , y ) = y 2 − x 2 , and y ( 0 ) = y 0 = 1

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 9, Problem 4BM

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To obtain: The approximation for the differential equation y′=y2−x2 for n = 5 with the initial condition y(0)=1, at x=0.5.

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Chapter 9, Problem 3BM

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Problem 11 (a) A tank is discharging water through an orifice at a depth of T meter below the surface of the water whose area is A m². The following are the values of a for the corresponding values of A: A 1.257 1.390 x 1.50 1.65 1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 Using the formula -3.0 (0.018)T = dx. calculate T, the time in seconds for the level of the water to drop from 3.0 m to 1.5 m above the orifice. (b) The velocity of a train which starts from rest is given by the fol- lowing table, the time being reckoned in minutes from the start and the speed in km/hour: | † (minutes) |2|4 6 8 10 12 14 16 18 20 v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 Estimate approximately the total distance ran in 20 minutes.

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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To obtain: The approximation for the differential equation y ′ = y 2 − x 2 for n = 5 with the initial condition y ( 0 ) = 1 , at x = 0.5 .

Solution Summary: The author explains Euler's method for obtaining the approximation for the differential equation yprime, where y is a number greater than x_0 and

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To obtain: The approximation for the differential equation y′=y2−x2 for n = 5 with the initial condition y(0)=1, at x=0.5.

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Chapter 9 Solutions