Fundamentals of Differential Equations and Boundary Value Problems
Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
bartleby

Concept explainers

bartleby

Videos

Textbook Question
Book Icon
Chapter 3.7, Problem 1E

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem

y = cos ( x + y ) , y ( 0 ) = π .

Expert Solution & Answer
Check Mark
To determine

To find:

The recursive formulas for the Taylor method of order 2.for the differential equation y=cos(x+y),y(0)=π.

Answer to Problem 1E

Solution:

The recursive formulas for the Taylor method of order 2 is yn+1=yn+hcos(xn+yn)+h22sin(x+y)+sin(xn+yn)cos(xn+yn).

Explanation of Solution

Formula used:

The recursive formulas for Taylor method of order second is,

xn+1=xn+hyn+1=yn+hf(xn,yn)+h22f2(xn,yn)

Calculation:

Consider the differential equation.

y=cos(x+y) ……(1)

The differential equation is the function of x and y so,

y=f(x,y),y(x0)=y0 ……(2)

Compare equation (1) and equation (2).

f(x,y)=cos(x+y)x0=0yo=π

Differentiate the function f(x,y)=cos(x+y) with respect to x,

fx(x,y)=sin(x+y)

Differentiate the function f(x,y)=cos(x+y) with respect to x,

fy(x,y)=sin(x+y)

To calculate,

f2(x,y)=fx+fy(f(x,y)) ……(3)

Substitute sin(x+y) for fx, sin(x+y) for fy and cos(x+y) for f(x,y) in equation (3).

f2(x,y)=sin(x+y)+sin(x+y)cos(x+y)

Consider the recursive formulas for Taylor method of order second.

yn+1=yn+hf(xn,yn)+h22f2(xn,yn) ……(4).

Substitute cos(xn+yn) for f(xn,yn) and sin(x+y)+sin(x+y)cos(x+y) for f2(xn,yn) in equation (4).

yn+1=yn+hcos(xn+yn)+h22sin(x+y)+sin(xn+yn)cos(xn+yn)

Therefore, the recursive formulas for the Taylor method of order 2 is yn+1=yn+hcos(xn+yn)+h22sin(x+y)+sin(xn+yn)cos(xn+yn).

Conclusion:

Thus, the recursive formulas for the Taylor method of order 2 is yn+1=yn+hcos(xn+yn)+h22sin(x+y)+sin(xn+yn)cos(xn+yn).

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
(y² + y) Use the following methods to approximate y(1.2) for the initial value problem 1sts1.2, y(1) = -2 and h = 0.1. (a) Euler method (b) Taylor method of order 2 (c) Runge-Kutta method of order 4
2. (a) Find the fourth-order Taylor polynomial Pa(x), and the remainder term of the function f (x) = cos [T (x + 1)] about xo -1. (b) Find the value of x that can be used in P4(x) in part (i) to approximate the value of cos (). (c) Use part (i) and (ii) to find an approximate value of cos find the absolute error if the true value is 3. Hence, NOTE: Take ™
4. Find the first three nonzero terms in the Taylor polynomial approximation for the first order problem y' = y?, y(0) = 2.

Chapter 3 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 3.2 - Prob. 11ECh. 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. 14ECh. 3.2 - Prob. 15ECh. 3.2 - 16 Show that for a differentiable function p(t),...Ch. 3.2 - Prob. 18ECh. 3.2 - Prob. 19ECh. 3.2 - Prob. 20ECh. 3.2 - A snowball melts in such a way that the rate of...Ch. 3.2 - Prob. 22ECh. 3.2 - Prob. 23ECh. 3.2 - Prob. 24ECh. 3.2 - Prob. 25ECh. 3.2 - Prob. 26ECh. 3.2 - Prob. 27ECh. 3.3 - Prob. 1ECh. 3.3 - Prob. 2ECh. 3.3 - Prob. 3ECh. 3.3 - Prob. 4ECh. 3.3 - Prob. 5ECh. 3.3 - Prob. 6ECh. 3.3 - Prob. 7ECh. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Early Monday morning, the temperature in the...Ch. 3.3 - During the summer the temperature inside a van...Ch. 3.3 - Prob. 12ECh. 3.3 - Prob. 13ECh. 3.3 - Prob. 14ECh. 3.3 - Prob. 15ECh. 3.3 - Prob. 16ECh. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - Prob. 3ECh. 3.4 - Prob. 4ECh. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 7ECh. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 9ECh. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.4 - Prob. 13ECh. 3.4 - Prob. 14ECh. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.4 - In Problem 16, let I=50 kg-m2 and the retarding...Ch. 3.4 - Prob. 18ECh. 3.4 - Prob. 19ECh. 3.4 - Prob. 20ECh. 3.4 - Prob. 21ECh. 3.4 - Prob. 22ECh. 3.4 - Prob. 23ECh. 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. 2ECh. 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. 5ECh. 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. 1ECh. 3.6 - Prob. 2ECh. 3.6 - Prob. 3ECh. 3.6 - In Example 1, page 126, the improved Eulers method...Ch. 3.6 - Prob. 5ECh. 3.6 - Prob. 6ECh. 3.6 - Prob. 7ECh. 3.6 - Use the improved Eulers method subroutine with...Ch. 3.6 - Prob. 9ECh. 3.6 - Prob. 10ECh. 3.6 - Use the improved Eulers method with tolerance to...Ch. 3.6 - Prob. 12ECh. 3.6 - Prob. 13ECh. 3.6 - Prob. 14ECh. 3.6 - The solution to the initial value problem...Ch. 3.6 - Prob. 16ECh. 3.6 - Prob. 17ECh. 3.6 - Prob. 18ECh. 3.6 - Prob. 20ECh. 3.7 - Determine the recursive formulas for the Taylor...Ch. 3.7 - Determine the recursive formulas for the Taylor...Ch. 3.7 - Prob. 3ECh. 3.7 - Prob. 4ECh. 3.7 - Prob. 5ECh. 3.7 - Prob. 6ECh. 3.7 - Prob. 7ECh. 3.7 - Prob. 8ECh. 3.7 - Prob. 9ECh. 3.7 - Prob. 10ECh. 3.7 - Prob. 11ECh. 3.7 - Prob. 12ECh. 3.7 - Prob. 13ECh. 3.7 - Prob. 14ECh. 3.7 - Prob. 15ECh. 3.7 - Prob. 16ECh. 3.7 - The Taylor method of order 2 can be used to...Ch. 3.7 - Prob. 18ECh. 3.7 - Prob. 19ECh. 3.7 - Prob. 20ECh. 3.7 - Prob. 21E
Knowledge Booster
Background pattern image
Math
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.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY
Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY