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We consider the initial value problem 18z'y" – 21zy' + 6y = 0, y(1) = 2, y'(1) = -2 By looking for solutions in the form y = z' in an Euler-Cauchy problem Az'y" + Bry' + Cy = 0, we obtain a auxiliary equation Ar? + (B – A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: - 0 (2) Find the roots of the auxiliary equation: (enter your results as a comma separated list ) (3) Find a fundamental set of solutions y1, 2: (enter your results as a comma separated list) (4) Recall that the complementary solution (i.e., the general solution) is ye = ciyı + c2y2. Find the unique solution satisfying y(1) = 2, y'(1) = -2

We consider the initial value problem 18z'y" – 21zy' + 6y = 0, y(1) = 2, y'(1) = -2 By looking for solutions in the form y = z' in an Euler-Cauchy problem Az'y" + Bry' + Cy = 0, we obtain a auxiliary equation Ar? + (B – A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: - 0 (2) Find the roots of the auxiliary equation: (enter your results as a comma separated list ) (3) Find a fundamental set of solutions y1, 2: (enter your results as a comma separated list) (4) Recall that the complementary solution (i.e., the general solution) is ye = ciyı + c2y2. Find the unique solution satisfying y(1) = 2, y'(1) = -2

Calculus: Early Transcendentals
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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We consider the initial value problem \(18x^2y'' - 21xy' + 6y = 0, \quad y(1) = 2, \quad y'(1) = -2\). By looking for solutions in the form \(y = x^r\) in an Euler-Cauchy problem \(Ax^2y'' + Bxy' + Cy = 0\), we obtain an auxiliary equation \(Ar^2 + (B - A)r + C = 0\) which is the analog of the auxiliary equation in the constant coefficient case. 1. For this problem find the auxiliary equation: \[ \_\_\_ = 0\] 2. Find the roots of the auxiliary equation: (enter your results as a comma-separated list) 3. Find a fundamental set of solutions \(y_1, y_2:\) (enter your results as a comma-separated list) 4. Recall that the complementary solution (i.e., the general solution) is \(y_c = c_1y_1 + c_2y_2\). Find the unique solution satisfying \(y(1) = 2, \quad y'(1) = -2\).
Transcribed Image Text:We consider the initial value problem \(18x^2y'' - 21xy' + 6y = 0, \quad y(1) = 2, \quad y'(1) = -2\). By looking for solutions in the form \(y = x^r\) in an Euler-Cauchy problem \(Ax^2y'' + Bxy' + Cy = 0\), we obtain an auxiliary equation \(Ar^2 + (B - A)r + C = 0\) which is the analog of the auxiliary equation in the constant coefficient case. 1. For this problem find the auxiliary equation: \[ \_\_\_ = 0\] 2. Find the roots of the auxiliary equation: (enter your results as a comma-separated list) 3. Find a fundamental set of solutions \(y_1, y_2:\) (enter your results as a comma-separated list) 4. Recall that the complementary solution (i.e., the general solution) is \(y_c = c_1y_1 + c_2y_2\). Find the unique solution satisfying \(y(1) = 2, \quad y'(1) = -2\).
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