We consider the initial value problem zy" – 7zy' + 16y = 0, y(1) = -2, y'(1) = -9 By looking for solutions in the formy = 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: (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) = -9
We consider the initial value problem zy" – 7zy' + 16y = 0, y(1) = -2, y'(1) = -9 By looking for solutions in the formy = 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: (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) = -9
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 \( x^2y'' - 7xy' + 16y = 0, \, y(1) = -2, \, y'(1) = -9 \).
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: [Input box] = 0
2. Find the roots of the auxiliary equation: [Input box] (enter your results as a comma-separated list)
3. Find a fundamental set of solutions \( y_1, y_2 \): [Input box] (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 \( y(1) = -2, \, y'(1) = -9 \)
\[ y = \] [Input box]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8edd889e-338a-4275-872c-5b2b400b4e22%2Fde55cf53-e70b-46f2-a930-d4c882b8b6fc%2Fwr1enae_processed.png&w=3840&q=75)
Transcribed Image Text:We consider the initial value problem \( x^2y'' - 7xy' + 16y = 0, \, y(1) = -2, \, y'(1) = -9 \).
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: [Input box] = 0
2. Find the roots of the auxiliary equation: [Input box] (enter your results as a comma-separated list)
3. Find a fundamental set of solutions \( y_1, y_2 \): [Input box] (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 \( y(1) = -2, \, y'(1) = -9 \)
\[ y = \] [Input box]
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