Consider the differential equation 12x²y" + 23xy' + (12x² - 1) y = 0. Determine the indicial equation using the variable r. 12 r² + 11 r −1 0 ✓ Find the recurrence relation. (-1)-1.12n-1 an i = (r+n) (12 (r+ n − 1 +23) − 1) - Find the roots of the indicial relation., Number of roots: two 71 = X DO n=1 1 12 an-2 for n = 2, 3, ... 72 = Find the series solution (x > 0) corresponding to the larger root. (-1)"62 y₁(x) = x21+ n! 37. 61 (24n+13) -1 ✓
Consider the differential equation 12x²y" + 23xy' + (12x² - 1) y = 0. Determine the indicial equation using the variable r. 12 r² + 11 r −1 0 ✓ Find the recurrence relation. (-1)-1.12n-1 an i = (r+n) (12 (r+ n − 1 +23) − 1) - Find the roots of the indicial relation., Number of roots: two 71 = X DO n=1 1 12 an-2 for n = 2, 3, ... 72 = Find the series solution (x > 0) corresponding to the larger root. (-1)"62 y₁(x) = x21+ n! 37. 61 (24n+13) -1 ✓
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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![### Solving a Differential Equation Using the Frobenius Method
#### Given Differential Equation
\[ 12x^2y'' + 23xy' + (12x^2 - 1)y = 0 \]
#### Step 1: Determine the Indicial Equation
Using the variable \( r \), the indicial equation is found as follows:
\[ 12r^2 + 11r - 1 = 0 \]
#### Step 2: Find the Recurrence Relation
To determine the recurrence relation, use:
\[ a_n = \frac{(-1)^{n-1} \cdot 12^{n-1}}{(r+n)(12(r+n-1+23)-1)} \]
This should give:
\[ a_{n-2} \text{ for } n = 2, 3, \ldots \]
#### Step 3: Find the Roots of the Indicial Relation
Solving the indicial equation \( 12r^2 + 11r - 1 = 0 \) gives us:
- Number of roots: Two
- Roots are:
\[ r_1 = \frac{1}{12} \]
\[ r_2 = -1 \]
#### Step 4: Find the Series Solution \( (x > 0) \) Corresponding to the Larger Root
The series solution can be expressed as:
\[ y_1(x) = x^{r_1} \left[ 1 + \sum_{n=1}^{\infty} \frac{(-1)^n 6^n x^{2n}}{n! \cdot 37 \cdot 61 \cdot (24n+13)} \right] \]
### Summary
This step-by-step process outlines the application of the Frobenius method to find the series solution to the given differential equation while considering the indicial equation, recurrence relation, and roots of the indicial relation. The final series solution corresponds to the larger root found.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2F8d689af3-4962-4187-b3b2-431414153e77%2Fctjuzh5m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a Differential Equation Using the Frobenius Method
#### Given Differential Equation
\[ 12x^2y'' + 23xy' + (12x^2 - 1)y = 0 \]
#### Step 1: Determine the Indicial Equation
Using the variable \( r \), the indicial equation is found as follows:
\[ 12r^2 + 11r - 1 = 0 \]
#### Step 2: Find the Recurrence Relation
To determine the recurrence relation, use:
\[ a_n = \frac{(-1)^{n-1} \cdot 12^{n-1}}{(r+n)(12(r+n-1+23)-1)} \]
This should give:
\[ a_{n-2} \text{ for } n = 2, 3, \ldots \]
#### Step 3: Find the Roots of the Indicial Relation
Solving the indicial equation \( 12r^2 + 11r - 1 = 0 \) gives us:
- Number of roots: Two
- Roots are:
\[ r_1 = \frac{1}{12} \]
\[ r_2 = -1 \]
#### Step 4: Find the Series Solution \( (x > 0) \) Corresponding to the Larger Root
The series solution can be expressed as:
\[ y_1(x) = x^{r_1} \left[ 1 + \sum_{n=1}^{\infty} \frac{(-1)^n 6^n x^{2n}}{n! \cdot 37 \cdot 61 \cdot (24n+13)} \right] \]
### Summary
This step-by-step process outlines the application of the Frobenius method to find the series solution to the given differential equation while considering the indicial equation, recurrence relation, and roots of the indicial relation. The final series solution corresponds to the larger root found.
![### Differential Equations: Identifying Regular Singular Points and Solving Indicial Equations
**Problem Statement:**
Show that the given differential equation \(3x^2y'' + \left(2x^2 + \frac{3}{4}\right)y = 0\) has a regular singular point at \(x=0\).
\[ xp(x) = \]
\[ x^2 q(x) = \]
---
**Find Indicial Equation:**
\[0 = \]
---
**How many roots does the indicial equation have?**
[Choose one ▼]
---
**Find the Recurrence Equation:**
\[a_n = \]
[Choose one ▼]
---
**Explanation of Steps and Concepts:**
1. **Identifying a Regular Singular Point:**
- For a differential equation of the form \(a_2(x)y'' + a_1(x)y' + a_0(x)y = 0\), if \(a_2(0)=0\), this implies a singular point. To check if it is regular, \( \frac{a_1(x)}{a_2(x)}\) and \( \frac{a_0(x)}{a_2(x)}\) must be analytic at \(x=0\).
2. **Verifying the Regular Singular Point:**
- Here, \(a_2(x) = 3x^2\), \(a_1(x) = 0\), and \(a_0(x) = \left(2x^2 + \frac{3}{4}\right)\).
- We need to express \(p(x)\) and \(q(x)\):
\[
p(x) = \frac{a_1(x)}{a_2(x)} = \frac{0}{3x^2} = 0
\]
\[
q(x) = \frac{a_0(x)}{a_2(x)} = \frac{2x^2 + \frac{3}{4}}{3x^2}
\]
- Substitute in the respective boxes.
3. **Finding the Indicial Equation:**
- By substituting the Frobenius series \(y = \sum_{n=0}^{\infty} a_n x^{n+r}\) into the differential equation and equating coefficients of like powers of \(x\), we](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20c5837f-8a07-4fb0-b110-a8d2b8a51a74%2F8d689af3-4962-4187-b3b2-431414153e77%2F6111n6r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Identifying Regular Singular Points and Solving Indicial Equations
**Problem Statement:**
Show that the given differential equation \(3x^2y'' + \left(2x^2 + \frac{3}{4}\right)y = 0\) has a regular singular point at \(x=0\).
\[ xp(x) = \]
\[ x^2 q(x) = \]
---
**Find Indicial Equation:**
\[0 = \]
---
**How many roots does the indicial equation have?**
[Choose one ▼]
---
**Find the Recurrence Equation:**
\[a_n = \]
[Choose one ▼]
---
**Explanation of Steps and Concepts:**
1. **Identifying a Regular Singular Point:**
- For a differential equation of the form \(a_2(x)y'' + a_1(x)y' + a_0(x)y = 0\), if \(a_2(0)=0\), this implies a singular point. To check if it is regular, \( \frac{a_1(x)}{a_2(x)}\) and \( \frac{a_0(x)}{a_2(x)}\) must be analytic at \(x=0\).
2. **Verifying the Regular Singular Point:**
- Here, \(a_2(x) = 3x^2\), \(a_1(x) = 0\), and \(a_0(x) = \left(2x^2 + \frac{3}{4}\right)\).
- We need to express \(p(x)\) and \(q(x)\):
\[
p(x) = \frac{a_1(x)}{a_2(x)} = \frac{0}{3x^2} = 0
\]
\[
q(x) = \frac{a_0(x)}{a_2(x)} = \frac{2x^2 + \frac{3}{4}}{3x^2}
\]
- Substitute in the respective boxes.
3. **Finding the Indicial Equation:**
- By substituting the Frobenius series \(y = \sum_{n=0}^{\infty} a_n x^{n+r}\) into the differential equation and equating coefficients of like powers of \(x\), we
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