4. Suppose the characteristic equation of a second-order Cauchy-Euler equation has a repeated root r = c E R. (a) What's one solution of this ODE? Call this y1. (b) To find a second solution that is linearly independent from the first, suppose y2 = u · Y1, where u = u(x). Solve for u(x) and then find y2(x). (c) Show that yı and y2 are indeed linearly independent. 5. (a) Rewrite ta+Bi using Euler's Formula. (b) Construct an example of a Cauchy-Euler equation whose characteristic equation has non- real roots. (c) Solve the equation you constructed in part (b) and use the formula from part (a) to rewrite the solution in terms of real valued functions.
4. Suppose the characteristic equation of a second-order Cauchy-Euler equation has a repeated root r = c E R. (a) What's one solution of this ODE? Call this y1. (b) To find a second solution that is linearly independent from the first, suppose y2 = u · Y1, where u = u(x). Solve for u(x) and then find y2(x). (c) Show that yı and y2 are indeed linearly independent. 5. (a) Rewrite ta+Bi using Euler's Formula. (b) Construct an example of a Cauchy-Euler equation whose characteristic equation has non- real roots. (c) Solve the equation you constructed in part (b) and use the formula from part (a) to rewrite the solution in terms of real valued functions.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![4. Suppose the characteristic equation of a second-order Cauchy-Euler equation has a repeated
root r = c E R.
(a) What's one solution of this ODE? Call this y1.
(b) To find a second solution that is linearly independent from the first, suppose y2 = u · Y1,
where u = u(x). Solve for u(x) and then find Y2(x).
(c) Show that Yı and y2 are indeed linearly independent.
5. (a) Rewrite ta+ßi using Euler's Formula.
(b) Construct an example of a Cauchy-Euler equation whose characteristic equation has non-
real roots.
(c) Solve the equation you constructed in part (b) and use the formula from part (a) to
rewrite the solution in terms of real valued functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F234a7be1-58c3-4913-b976-8fa32064a420%2Ff6637306-2f50-4c5e-9f55-21f2880e8445%2Fstfu0o_processed.png&w=3840&q=75)
Transcribed Image Text:4. Suppose the characteristic equation of a second-order Cauchy-Euler equation has a repeated
root r = c E R.
(a) What's one solution of this ODE? Call this y1.
(b) To find a second solution that is linearly independent from the first, suppose y2 = u · Y1,
where u = u(x). Solve for u(x) and then find Y2(x).
(c) Show that Yı and y2 are indeed linearly independent.
5. (a) Rewrite ta+ßi using Euler's Formula.
(b) Construct an example of a Cauchy-Euler equation whose characteristic equation has non-
real roots.
(c) Solve the equation you constructed in part (b) and use the formula from part (a) to
rewrite the solution in terms of real valued functions.
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