Explanation Given: The given differential equation is, y ′ = x + 1 − y y ( 0 ) = 1 The actual solution is, y = x + e − x Formula used: The recursive formulas for Taylor method of order second are, x n + 1 = x n + h y n + 1 = y n + h f ( x n , y n ) + h 2 2 f 2 ( x n , y n ) The recursive formulas for Taylor method of order fourth are, x n + 1 = x n + h y n + 1 = y n + h f ( x n , y n ) + h 2 2 f 2 ( x n , y n ) + h 3 3 ! f 3 ( x n , y n ) + h 4 4 ! f 4 ( x n , y n ) Calculation: Consider the differential equation. y ′ = x + 1 − y ... ( 1 ) The differential equation is the function of x and y so, y ′ = f ( x , y ) , y ( x 0 ) = y 0 ... ( 2 ) Compare equation ( 1 ) and equation ( 2 ) . f ( x , y ) = x + 1 − y x 0 = 0 y o = 1 Differentiate the function f ( x , y ) = x + 1 − y with respect to x , f x ( x , y ) = 1 Differentiate the function f ( x , y ) = x + 1 − y with respect to x , f y ( x , y ) = − 1 To calculate, f 2 ( x , y ) = f x + f y ( f ( x , y ) ) ... ( 3 ) Substitute 1 for f x , − 1 for f y and x − y for f ( x , y ) in equation ( 3 ) . f 2 ( x , y ) = y − x Consider the recursive formulas for Taylor method of order two. y n + 1 = y n + h f ( x n , y n ) + h 2 2 f 2 ( x n , y n ) ... ( 4 ) Substitute x n + 1 − y n for f ( x n , y n ) , y n − x n for f 2 ( x n , y n ) in equation ( 4 ) . y n + 1 = y n + h ( x n + 1 − y n ) + h 2 2 ( y n − x n ) ... ( 5 ) Consider the recursive formulas for Taylor method of order four. y n + 1 = y n + h f ( x n , y n ) + h 2 2 f 2 ( x n , y n ) + h 3 3 ! f 3 ( x n , y n ) + h 4 4 ! f 4 ( x n , y n ) ... ( 6 ) f 3 ( x , y ) = y ‴ = d f 2 d x ... ( 7 ) f 4 ( x , y ) = d f 3 d x ... ( 8 ) Differentiate equation ( 7 ) with respect to x . f 3 ( x , y ) = d d x ( y − x ) = d y d x − 1 = x − y Differentiate equation ( 8 ) with respect to x . f 4 ( x , y ) = d d x ( x − y ) = 1 − d y d x = y − x Substitute x n + 1 − y n for f ( x n , y n ) , y n − x n for f 2 ( x n , y n ) , x n − y n for f 3 ( x n , y n ) and y n − x n for f 4 ( x n , y n ) in equation ( 6 )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 3.7, Problem 5E

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Saved Tempo Company's fixed budget (based on sales of 18,000 units) folllows Fixed Budget Sales (18,000 units x $201 per unit) 3,618,000 Costs Direct materials Direct labor Indirect materials Supervisor salary 432,000 792,000 486,000 232,000 Sales commissions 126,000 Shipping 270,000 Administrative salaries 232,000 Depreciation-office equipment 252,000 Insurance 222,000 Office rent 232,000 Income 292,000 1. Compute total variable cost per unit. 2. Compute total fixed costs 3. Prepare a flexible budget at activity levels of 16,000 units and 20,000 units. Complete this question by entering your answers in the tabs below. Q Search hp PRES 0 O

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Fundamentals of Differential Equations and Boundary Value Problems

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The solution for the Taylor method of order 2 and order 4 .

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The solution for the Taylor method of order 2 and order 4 .

Solution Summary: The author explains the recursive formulas for Taylor method of order two and order four.

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The solution for the Taylor method of order 2 and order 4.