2. Consider the DE x²y" + xy' + y = 0, x > 0. (A) [10%] Show that the verification of the trial function y(x) = x as a solution of the DE leads to λ = ±i. (B) [10%] Using differentiation, show that y₁(x) = cos(ln x) is a solution of the DE. (C) [80%] Using the reduction of order method, find the second linearly independent solution of the DE. Then, find the solution that is subject to the initial conditions y(1) = 0 and y'(1) = 1.
2. Consider the DE x²y" + xy' + y = 0, x > 0. (A) [10%] Show that the verification of the trial function y(x) = x as a solution of the DE leads to λ = ±i. (B) [10%] Using differentiation, show that y₁(x) = cos(ln x) is a solution of the DE. (C) [80%] Using the reduction of order method, find the second linearly independent solution of the DE. Then, find the solution that is subject to the initial conditions y(1) = 0 and y'(1) = 1.
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...
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i need help please . and please dont use chat gpt i am trying to learn and see the mistake i did when solving mine
![2. Consider the DE
x²y" + xy' + y = 0,
x > 0.
(A) [10%] Show that the verification of the trial function y(x) = x as a solution of the DE leads
to λ = ±i.
(B) [10%] Using differentiation, show that y₁(x) = cos(ln x) is a solution of the DE.
(C) [80%] Using the reduction of order method, find the second linearly independent solution of
the DE. Then, find the solution that is subject to the initial conditions y(1) = 0 and y'(1) = 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86794e1d-a025-469e-9776-3458200614a7%2F3e8a3b50-9729-40d5-9241-a73ef752737c%2Fpaecxd_processed.png&w=3840&q=75)
Transcribed Image Text:2. Consider the DE
x²y" + xy' + y = 0,
x > 0.
(A) [10%] Show that the verification of the trial function y(x) = x as a solution of the DE leads
to λ = ±i.
(B) [10%] Using differentiation, show that y₁(x) = cos(ln x) is a solution of the DE.
(C) [80%] Using the reduction of order method, find the second linearly independent solution of
the DE. Then, find the solution that is subject to the initial conditions y(1) = 0 and y'(1) = 1.
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