Previously, we learned some of the many important applications of differential equations, and learned some approaches to solve or analyze them. Here, we consider an important approach that will allow us to solve a wider variety of differential equations. Let's consider the familiar differential equation from exponential population growth given by =ky, where k is the constant of proportionality. While we can solve this differential equation using familiar methods, we take a different approach now that can be applied to a much larger set of differential equations. For the rest of this activity, let's assume that k = 1. We will use our knowledge of Taylor series to find a solution to the differential equation. To do so, we assume that we have a solution y = f(x) and that f(x) has a Taylor series that can be written in the form ∞ y = f(x) = - Σαΐα, i=0 where the coefficients a; are undetermined. Our task is to find the coefficients.
Previously, we learned some of the many important applications of differential equations, and learned some approaches to solve or analyze them. Here, we consider an important approach that will allow us to solve a wider variety of differential equations. Let's consider the familiar differential equation from exponential population growth given by =ky, where k is the constant of proportionality. While we can solve this differential equation using familiar methods, we take a different approach now that can be applied to a much larger set of differential equations. For the rest of this activity, let's assume that k = 1. We will use our knowledge of Taylor series to find a solution to the differential equation. To do so, we assume that we have a solution y = f(x) and that f(x) has a Taylor series that can be written in the form ∞ y = f(x) = - Σαΐα, i=0 where the coefficients a; are undetermined. Our task is to find the coefficients.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Part (b)
Two series are equal if and only if they have the same coefficients on like power terms. Use this fact to write a₁ in terms of ao. When writing your answer, use a0 for að.
Answer: a1
Part (c)
Now write a2 in terms of a₁. Then write a2 in terms of ag. When writing your answer, use al for a₁
Answer in terms of a₁: a2
Answer in terms of ao: a2
Part (d)
Write ag in terms of a2. Then write ag in terms of ao. When writing your answer, use a2 for a2
Answer in terms of a2: a3
Answer in terms of an: a3
Transcribed Image Text:Previously, we learned some of the many important applications of differential equations, and learned some approaches to solve or analyze them. Here, we
consider an important approach that will allow us to solve a wider variety of differential equations.
Let's consider the familiar differential equation from exponential population growth given by
y = ky,
where k is the constant of proportionality. While we can solve this differential equation using familiar methods, we take a different approach now that can be
applied to a much larger set of differential equations. For the rest of this activity, let's assume that k = 1. We will use our knowledge of Taylor series to find a
solution to the differential equation.
To do so, we assume that we have a solution y = f(x) and that f(x) has a Taylor series that can be written in the form
∞
y=f(x) = Σa;x¹,
i=0
where the coefficients a; are undetermined. Our task is to find the coefficients.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
