Explanation Given: The given differential equation is, y ′ = 1 − y y ( 0 ) = 0 The actual solution is, y = 1 − 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 ′ = 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 ) = 1 − y x 0 = 0 y o = 0 Differentiate the function f ( x , y ) = 1 − y with respect to x , f x ( x , y ) = 0 Differentiate the function f ( x , y ) = 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 0 for f x , − 1 for f y and x − y for f ( x , y ) in equation ( 3 ) . f 2 ( x , y ) = y − 1 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 1 − y n for f ( x n , y n ) , y n − 1 for f 2 ( x n , y n ) in equation ( 4 ) . y n + 1 = y n + h ( 1 − y n ) + h 2 2 ( y n − 1 ) …… ( 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 + 1 ) = − y + 1 Differentiate equation ( 8 ) with respect to x . f 4 ( x , y ) = d d x ( 1 − y ) = y − 1 Substitute 1 − y n for f ( x n , y n ) , y n − 1 for f 2 ( x n , y n ) , 1 − y n for f 3 ( x n , y n ) and y n − 1 for f 4 ( x n , y n ) in equation ( 6 )

7th Edition

ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

See similar textbooks

Videos

Question

Chapter 3.7, Problem 6E

To determine

To find:

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.7, Problem 5E

Chapter 3.7, Problem 7E

Students have asked these similar questions

DO these math problems without ai, show the solutions as well. and how you solved it. and could you do it with in the time spand

The Cartesian coordinates of a point are given. (a) (-8, 8) (i) Find polar coordinates (r, 0) of the point, where r > 0 and 0 ≤ 0 0 and 0 ≤ 0 < 2π. (1, 0) = (r. = ([ (ii) Find polar coordinates (r, 8) of the point, where r < 0 and 0 ≤ 0 < 2π. (5, 6) = =([

The Cartesian coordinates of a point are given. (a) (4,-4) (i) Find polar coordinates (r, e) of the point, where r > 0 and 0 0 and 0 < 0 < 2π. (r, 6) = X 7 (ii) Find polar coordinates (r, 8) of the point, where r < 0 and 0 0 < 2π. (r, 0) = X

Chapter 3 Solutions

EBK FUND.OF DIFF.EQUATIONS+BOUNDARY...

Ch. 3.2 - A brine solution of salt flows at a constant rate...Ch. 3.2 - Prob. 2E Ch. 3.2 - Prob. 3E Ch. 3.2 - A brine solution of salt flows at a constant rate...Ch. 3.2 - A swimming pool whose volume is 10,000gal contains...Ch. 3.2 - The air in a small room 12ft by 8ft by 8ft is 3...Ch. 3.2 - Beginning at time t=0, fresh water is pumped at...Ch. 3.2 - A tank initially contains S0lb of salt dissolved...Ch. 3.2 - In 1990 the Department of Natural Resources...Ch. 3.2 - Prob. 10E

Ch. 3.2 - Prob. 11E Ch. 3.2 - For the logistic curve15, assume pa:=p(ta) and...Ch. 3.2 - In Problem 9, suppose we have the additional...Ch. 3.2 - Prob. 14E Ch. 3.2 - Prob. 15E Ch. 3.2 - 16 Show that for a differentiable function p(t),...Ch. 3.2 - Prob. 18E Ch. 3.2 - Prob. 19E Ch. 3.2 - Prob. 20E Ch. 3.2 - A snowball melts in such a way that the rate of...Ch. 3.2 - Prob. 22E Ch. 3.2 - Prob. 23E Ch. 3.2 - Prob. 24E Ch. 3.2 - Prob. 25E Ch. 3.2 - Prob. 26E Ch. 3.2 - Prob. 27E Ch. 3.3 - Prob. 1E Ch. 3.3 - Prob. 2E Ch. 3.3 - Prob. 3E Ch. 3.3 - Prob. 4E Ch. 3.3 - Prob. 5E Ch. 3.3 - Prob. 6E Ch. 3.3 - Prob. 7E Ch. 3.3 - Prob. 8E Ch. 3.3 - Prob. 9E Ch. 3.3 - Early Monday morning, the temperature in the...Ch. 3.3 - During the summer the temperature inside a van...Ch. 3.3 - Prob. 12E Ch. 3.3 - Prob. 13E Ch. 3.3 - Prob. 14E Ch. 3.3 - Prob. 15E Ch. 3.3 - Prob. 16E Ch. 3.4 - Prob. 1E Ch. 3.4 - Prob. 2E Ch. 3.4 - Prob. 3E Ch. 3.4 - Prob. 4E Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 7E Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 9E Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.4 - Prob. 13E Ch. 3.4 - Prob. 14E Ch. 3.4 - Prob. 15E Ch. 3.4 - Prob. 16E Ch. 3.4 - In Problem 16, let I=50 kg-m2 and the retarding...Ch. 3.4 - Prob. 18E Ch. 3.4 - Prob. 19E Ch. 3.4 - Prob. 20E Ch. 3.4 - Prob. 21E Ch. 3.4 - Prob. 22E Ch. 3.4 - Prob. 23E Ch. 3.4 - Rocket Flight. A model rocket having initial mass...Ch. 3.4 - Escape Velocity. According to Newtons law of...Ch. 3.5 - An RL circuit with a 5- resistor and a 0.05-H...Ch. 3.5 - Prob. 2E Ch. 3.5 - The pathway for a binary electrical signal between...Ch. 3.5 - If the resistance in the RL circuit of Figure...Ch. 3.5 - Prob. 5E Ch. 3.5 - 6. Derive a power balance equation for the RL and...Ch. 3.5 - 7. An industrial electromagnet can be modeled as...Ch. 3.5 - 8. A 108F capacitor 10 nanofarads is charged to 50...Ch. 3.6 - Prob. 1E Ch. 3.6 - Prob. 2E Ch. 3.6 - Prob. 3E Ch. 3.6 - In Example 1, page 126, the improved Eulers method...Ch. 3.6 - Prob. 5E Ch. 3.6 - Prob. 6E Ch. 3.6 - Prob. 7E Ch. 3.6 - Use the improved Eulers method subroutine with...Ch. 3.6 - Prob. 9E Ch. 3.6 - Prob. 10E Ch. 3.6 - Use the improved Eulers method with tolerance to...Ch. 3.6 - Prob. 12E Ch. 3.6 - Prob. 13E Ch. 3.6 - Prob. 14E Ch. 3.6 - The solution to the initial value problem...Ch. 3.6 - Prob. 16E Ch. 3.6 - Prob. 17E Ch. 3.6 - Prob. 18E Ch. 3.6 - Prob. 20E Ch. 3.7 - Determine the recursive formulas for the Taylor...Ch. 3.7 - Determine the recursive formulas for the Taylor...Ch. 3.7 - Prob. 3E Ch. 3.7 - Prob. 4E Ch. 3.7 - Prob. 5E Ch. 3.7 - Prob. 6E Ch. 3.7 - Prob. 7E Ch. 3.7 - Prob. 8E Ch. 3.7 - Prob. 9E Ch. 3.7 - Prob. 10E Ch. 3.7 - Prob. 11E Ch. 3.7 - Prob. 12E Ch. 3.7 - Prob. 13E Ch. 3.7 - Prob. 14E Ch. 3.7 - Prob. 15E Ch. 3.7 - Prob. 16E Ch. 3.7 - The Taylor method of order 2 can be used to...Ch. 3.7 - Prob. 18E Ch. 3.7 - Prob. 19E Ch. 3.7 - Prob. 20E Ch. 3.7 - Prob. 21E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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

Solution Summary: The author explains that the solution for Taylor method of order 2 and order 4 is y(1)=0.62747.

Videos

To find:

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

Want to see the full answer?

Chapter 3 Solutions

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

Solution Summary: The author explains that the solution for Taylor method of order 2 and order 4 is y(1)=0.62747.

Videos

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

Want to see the full answer?

Chapter 3 Solutions

To find:

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