Explanation Given: The initial value problem is y ″ − 4 x y ′ + 5 y = 0 ; y ( 0 ) = − 1 , y ′ ( 0 ) = 1 Approach: Rewrite the given equation a 2 ( x ) y ″ + a 1 ( x ) y ′ + a 0 ( x ) y = 0 in the form y ″ + p ( x ) y ′ + q ( x ) y = 0 Here, p ( x ) = a 1 ( x ) a 2 ( x ) and q ( x ) = a 0 ( x ) a 2 ( x ) If x = 0 is an ordinary point for this equation then, the general solution of the given equation is y ( x ) = ∑ n = 0 ∞ a n x n . Calculation: The initial value problem is y ″ − 4 x y ′ + 5 y = 0 … ( 1 ) Because p ( x ) = − 4 x and q ( x ) = 5 are analytic at x = 0 therefore, write the general solution of the given problem as y ( x ) = ∑ n = 0 ∞ a n x n This gives y ′ ( x ) = ∑ n = 1 ∞ n a n x n − 1 y ″ ( x ) = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 Substitute these values in equation ( 1 ) ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 − 4 x ∑ n = 1 ∞ n a n x n − 1 + 5 ∑ n = 0 ∞ a n x n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 − 4 ∑ n = 1 ∞ n a n x n + 5 ∑ n = 0 ∞ a n x n = 0 … ( 2 ) In order to make each term in the above equation like term, put k = n − 2 in the first summation, k = n in the second summation, k = n in the third summation Then, equation ( 2 ) becomes ∑ k = 0 ∞ ( k + 1 ) ( k + 2 ) a k + 2 x k − 4 ∑ k = 1 ∞ k a k x k + 5 ∑ k = 0 ∞ a k x k = 0 2 a 2 + ∑ k = 1 ∞ ( k + 1 ) ( k + 2 ) a k + 2 x k − 4 ∑ k = 1 ∞ k a k x k + 5 a 0 + 5 ∑ k = 1 ∞ a k x k = 0 ( 2 a 2 + 5 a 0 ) + ∑ k = 1 ∞ [ ( k + 1 ) ( k + 2 ) a k + 2 − 4 k a k + 5 a k ] x k = 0 This will make all the coefficients equal to zero

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Videos

Question

Chapter 8.3, Problem 30E

To determine

To find:

The cubic polynomial approximation for the solution to the given problem and plot the graph of the linear, quadratic and cubic polynomial.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 29E

Chapter 8.3, Problem 31E

Students have asked these similar questions

please help me with this question thanks guys

please help me solve

To: [Boss's Name] From: Nathaniel D Sain Date: 4/5/2025 Subject: Decision Analysis for Business Scenario Introduction to the Business Scenario Our delivery services business has been experiencing steady growth, leading to an increased demand for faster and more efficient deliveries. To meet this demand, we must decide on the best strategy to expand our fleet. The three possible alternatives under consideration are purchasing new delivery vehicles, leasing vehicles, or partnering with third-party drivers. The decision must account for various external factors, including fuel price fluctuations, demand stability, and competition growth, which we categorize as the states of nature. Each alternative presents unique advantages and challenges, and our goal is to select the most viable option using a structured decision-making approach. Alternatives and States of Nature The three alternatives for fleet expansion were chosen based on their cost implications, operational efficiency, and…

Chapter 8 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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 cubic polynomial approximation for the solution to the given problem and plot the graph of the linear, quadratic and cubic polynomial.

Solution Summary: The author explains how to find the cubic polynomial approximation for the solution to the given problem.

Videos

To find: The cubic polynomial approximation for the solution to the given problem and plot the graph of the linear, quadratic and cubic polynomial.

Want to see the full answer?

Chapter 8 Solutions

To find:

The cubic polynomial approximation for the solution to the given problem and plot the graph of the linear, quadratic and cubic polynomial.