We consider the initial value problem z?y" – 3zy + 4y = 0, y(1) = 1, y'(1) = -1 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 = Ciy1 + c2y2. Find the unique solution satisfying y(1) = 1, y(1) = -1
We consider the initial value problem z?y" – 3zy + 4y = 0, y(1) = 1, y'(1) = -1 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 = Ciy1 + c2y2. Find the unique solution satisfying y(1) = 1, 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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![We consider the initial value problem \( x^2 y'' - 3xy' + 4y = 0 \), \( y(1) = 1 \), \( y'(1) = -1 \).
By looking for solutions in the form \( y = x^r \) in an Euler-Cauchy problem \( Ax^2 y'' + Bx y' + Cy = 0 \), 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 \( y_c = c_1 y_1 + c_2 y_2 \). Find the unique solution satisfying \( y(1) = 1 \), \( y'(1) = -1 \):**
\[ y = \_\_\_\_\_ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8edd889e-338a-4275-872c-5b2b400b4e22%2F214a9222-9d30-4caa-b0f0-10bf2471389c%2Fjlr9ytf_processed.png&w=3840&q=75)
Transcribed Image Text:We consider the initial value problem \( x^2 y'' - 3xy' + 4y = 0 \), \( y(1) = 1 \), \( y'(1) = -1 \).
By looking for solutions in the form \( y = x^r \) in an Euler-Cauchy problem \( Ax^2 y'' + Bx y' + Cy = 0 \), 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 \( y_c = c_1 y_1 + c_2 y_2 \). Find the unique solution satisfying \( y(1) = 1 \), \( y'(1) = -1 \):**
\[ y = \_\_\_\_\_ \]
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