2. (Finding a second independent solution of an Equi-Dimensional (or Cauchy-Euler) equation with a double real root) In physical applications, we often encounter the Equi-Dimensional (or Cauchy-Euler) equation: ax'y"(x) + bxy'(x) + cy(x) = 0 for constants a, b and c, which has solutions of the form y = x'. In the case x*y"(x) + (1– 2b)xy'(x) +b²y(x) = 0 for constant b, (2) we obtain only one value r = b by substitution, and the hunt is on for a second independent solution: a. Verify by substitution that y, = x° is a solution of the differential equation (2). Clearly state your conclusion. c. In order to obtain a second solution, assume that it is of the form y,«) = ux) y,) and determine the unknown function ux) by reduction of order. ux) = , so y,x) = d. Verify that y, and y, are linearly independent and state the interval of independence. Clearly state your conclusion.
2. (Finding a second independent solution of an Equi-Dimensional (or Cauchy-Euler) equation with a double real root) In physical applications, we often encounter the Equi-Dimensional (or Cauchy-Euler) equation: ax'y"(x) + bxy'(x) + cy(x) = 0 for constants a, b and c, which has solutions of the form y = x'. In the case x*y"(x) + (1– 2b)xy'(x) +b²y(x) = 0 for constant b, (2) we obtain only one value r = b by substitution, and the hunt is on for a second independent solution: a. Verify by substitution that y, = x° is a solution of the differential equation (2). Clearly state your conclusion. c. In order to obtain a second solution, assume that it is of the form y,«) = ux) y,) and determine the unknown function ux) by reduction of order. ux) = , so y,x) = d. Verify that y, and y, are linearly independent and state the interval of independence. Clearly state your conclusion.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Finding a Second Independent Solution of an Equi-Dimensional (or Cauchy-Euler) Equation with a Double Real Root
In physical applications, we often encounter the Equi-Dimensional (or Cauchy-Euler) equation:
\[ ax^2y''(x) + bxy'(x) + cy(x) = 0 \]
for constants \( a, b, \) and \( c \),
which has solutions of the form \( y = x^r \).
In the case
\[ x^2y''(x) + (1 - 2b)xy'(x) + b^2y(x) = 0 \quad \text{for constant } b, \]
we obtain only one value \( r = b \) by substitution, and the hunt is on for a second independent solution:
1. **Task a**: Verify by substitution that \( y_1 = x^b \) is a solution of the differential equation.
- **Action**: Clearly state your conclusion.
2. **Task b**: In order to obtain a second solution, assume that it is of the form \( y_2(x) = u(x) y_1(x) \) and determine the unknown function \( u(x) \) by reduction of order.
- **Given**:
\[
u(x) = \quad, \quad \text{so} \quad y_2(x) =
\]
3. **Task c**: Verify that \( y_1 \) and \( y_2 \) are linearly independent and state the interval of independence.
- **Action**: Clearly state your conclusion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc600d4db-1a2b-41cb-87e5-08053c31d1dc%2F52a38e06-dcc9-47f9-8e1c-de9b4f803ac7%2Fqaqy773_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding a Second Independent Solution of an Equi-Dimensional (or Cauchy-Euler) Equation with a Double Real Root
In physical applications, we often encounter the Equi-Dimensional (or Cauchy-Euler) equation:
\[ ax^2y''(x) + bxy'(x) + cy(x) = 0 \]
for constants \( a, b, \) and \( c \),
which has solutions of the form \( y = x^r \).
In the case
\[ x^2y''(x) + (1 - 2b)xy'(x) + b^2y(x) = 0 \quad \text{for constant } b, \]
we obtain only one value \( r = b \) by substitution, and the hunt is on for a second independent solution:
1. **Task a**: Verify by substitution that \( y_1 = x^b \) is a solution of the differential equation.
- **Action**: Clearly state your conclusion.
2. **Task b**: In order to obtain a second solution, assume that it is of the form \( y_2(x) = u(x) y_1(x) \) and determine the unknown function \( u(x) \) by reduction of order.
- **Given**:
\[
u(x) = \quad, \quad \text{so} \quad y_2(x) =
\]
3. **Task c**: Verify that \( y_1 \) and \( y_2 \) are linearly independent and state the interval of independence.
- **Action**: Clearly state your conclusion.
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