**Problem Statement:** 11. Suppose that the three numbers \( r_1, r_2, \) and \( r_3 \) are distinct. Show that the three functions \( \exp(r_1 x), \exp(r_2 x), \) and \( \exp(r_3 x) \) are linearly independent by showing that their Wronskian given below is nonzero for all \( x \). \[ W = \exp[(r_1 + r_2 + r_3)x] \cdot \begin{vmatrix} 1 & 1 & 1 \\ r_1 & r_2 & r_3 \\ r_1^2 & r_2^2 & r_3^2 \end{vmatrix} \] **General Wronskian:** If \( y_1, y_2, \ldots, y_n \) are \( n \) solutions of the homogeneous nth-order linear equation \[ y^{(n)} + p_1(x)y^{(n-1)} + \ldots + p_{n-1}(x)y' + p_n(x)y = 0 \] on an open interval \( I \), where each \( p_i \) is continuous, then the Wronskian is \[ W = W(y_1, y_2, \ldots, y_n). \] If \( y_1, y_2, \ldots, y_n \) are linearly independent, then \( (1) \) _______________________ at each point of \( I \). **Determinant Evaluation:** To simplify \( W \), evaluate the determinant. \[ W = \exp[(r_1 + r_2 + r_3)x] \cdot (\underline{\hspace{2cm}}) \] Can \[ [(r_1 + r_2 + r_3)x] \] be zero? - No [ ] - Yes [ ] The other factor involves the numbers \( r_1, r_2, \) and \( r_3 \). This can be further factored as a product of binomials. \[ W = \exp [(r_1 + r_2 + r_3)x] \cdot \underline{(\text{Factor completely.})} \] **Select Values:** For what values of \(
**Problem Statement:** 11. Suppose that the three numbers \( r_1, r_2, \) and \( r_3 \) are distinct. Show that the three functions \( \exp(r_1 x), \exp(r_2 x), \) and \( \exp(r_3 x) \) are linearly independent by showing that their Wronskian given below is nonzero for all \( x \). \[ W = \exp[(r_1 + r_2 + r_3)x] \cdot \begin{vmatrix} 1 & 1 & 1 \\ r_1 & r_2 & r_3 \\ r_1^2 & r_2^2 & r_3^2 \end{vmatrix} \] **General Wronskian:** If \( y_1, y_2, \ldots, y_n \) are \( n \) solutions of the homogeneous nth-order linear equation \[ y^{(n)} + p_1(x)y^{(n-1)} + \ldots + p_{n-1}(x)y' + p_n(x)y = 0 \] on an open interval \( I \), where each \( p_i \) is continuous, then the Wronskian is \[ W = W(y_1, y_2, \ldots, y_n). \] If \( y_1, y_2, \ldots, y_n \) are linearly independent, then \( (1) \) _______________________ at each point of \( I \). **Determinant Evaluation:** To simplify \( W \), evaluate the determinant. \[ W = \exp[(r_1 + r_2 + r_3)x] \cdot (\underline{\hspace{2cm}}) \] Can \[ [(r_1 + r_2 + r_3)x] \] be zero? - No [ ] - Yes [ ] The other factor involves the numbers \( r_1, r_2, \) and \( r_3 \). This can be further factored as a product of binomials. \[ W = \exp [(r_1 + r_2 + r_3)x] \cdot \underline{(\text{Factor completely.})} \] **Select Values:** For what values of \(
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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![**Problem Statement:**
11. Suppose that the three numbers \( r_1, r_2, \) and \( r_3 \) are distinct. Show that the three functions \( \exp(r_1 x), \exp(r_2 x), \) and \( \exp(r_3 x) \) are linearly independent by showing that their Wronskian given below is nonzero for all \( x \).
\[
W = \exp[(r_1 + r_2 + r_3)x] \cdot \begin{vmatrix}
1 & 1 & 1 \\
r_1 & r_2 & r_3 \\
r_1^2 & r_2^2 & r_3^2
\end{vmatrix}
\]
**General Wronskian:**
If \( y_1, y_2, \ldots, y_n \) are \( n \) solutions of the homogeneous nth-order linear equation
\[ y^{(n)} + p_1(x)y^{(n-1)} + \ldots + p_{n-1}(x)y' + p_n(x)y = 0 \]
on an open interval \( I \), where each \( p_i \) is continuous, then the Wronskian is
\[ W = W(y_1, y_2, \ldots, y_n). \]
If \( y_1, y_2, \ldots, y_n \) are linearly independent, then \( (1) \) _______________________ at each point of \( I \).
**Determinant Evaluation:**
To simplify \( W \), evaluate the determinant.
\[
W = \exp[(r_1 + r_2 + r_3)x] \cdot (\underline{\hspace{2cm}})
\]
Can \[ [(r_1 + r_2 + r_3)x] \] be zero?
- No [ ]
- Yes [ ]
The other factor involves the numbers \( r_1, r_2, \) and \( r_3 \). This can be further factored as a product of binomials.
\[
W = \exp [(r_1 + r_2 + r_3)x] \cdot \underline{(\text{Factor completely.})}
\]
**Select Values:**
For what values of \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e7b22d7-3558-4370-8e59-8331abbb5ed3%2F142024bc-c7fb-48d5-a832-cbf8ad6c62c0%2Fmip5mc.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
11. Suppose that the three numbers \( r_1, r_2, \) and \( r_3 \) are distinct. Show that the three functions \( \exp(r_1 x), \exp(r_2 x), \) and \( \exp(r_3 x) \) are linearly independent by showing that their Wronskian given below is nonzero for all \( x \).
\[
W = \exp[(r_1 + r_2 + r_3)x] \cdot \begin{vmatrix}
1 & 1 & 1 \\
r_1 & r_2 & r_3 \\
r_1^2 & r_2^2 & r_3^2
\end{vmatrix}
\]
**General Wronskian:**
If \( y_1, y_2, \ldots, y_n \) are \( n \) solutions of the homogeneous nth-order linear equation
\[ y^{(n)} + p_1(x)y^{(n-1)} + \ldots + p_{n-1}(x)y' + p_n(x)y = 0 \]
on an open interval \( I \), where each \( p_i \) is continuous, then the Wronskian is
\[ W = W(y_1, y_2, \ldots, y_n). \]
If \( y_1, y_2, \ldots, y_n \) are linearly independent, then \( (1) \) _______________________ at each point of \( I \).
**Determinant Evaluation:**
To simplify \( W \), evaluate the determinant.
\[
W = \exp[(r_1 + r_2 + r_3)x] \cdot (\underline{\hspace{2cm}})
\]
Can \[ [(r_1 + r_2 + r_3)x] \] be zero?
- No [ ]
- Yes [ ]
The other factor involves the numbers \( r_1, r_2, \) and \( r_3 \). This can be further factored as a product of binomials.
\[
W = \exp [(r_1 + r_2 + r_3)x] \cdot \underline{(\text{Factor completely.})}
\]
**Select Values:**
For what values of \(
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