### Finding a Second Independent Solution of an Equi-Dimensional (or Cauchy-Euler) Equation with a Double Real RootIn 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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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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

### 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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