Find the Wronskian of the set {sin 3x, cos 3x} and determine if it is linearly independent or linearly dependent. Find the Wronskian of the set {1- x, 1+x, 1-3x}
Find the Wronskian of the set {sin 3x, cos 3x} and determine if it is linearly independent or linearly dependent. Find the Wronskian of the set {1- x, 1+x, 1-3x}
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
1. **Wronskian of Trigonometric Functions**
Find the Wronskian of the set \(\{\sin 3x, \cos 3x\}\) and determine if it is linearly independent or linearly dependent.
2. **Wronskian of Polynomial Functions**
Find the Wronskian of the set \(\{1 - x, 1 + x, 1 - 3x\}\).
### Explanation
The Wronskian is a determinant used in the study of differential equations to determine whether a set of functions is linearly independent.
Let's analyze both sets:
#### 1. **\(\{\sin 3x, \cos 3x\}\):**
- The Wronskian \(W(f, g)\) of two functions \(f(x)\) and \(g(x)\), in this case, \(\sin 3x\) and \(\cos 3x\), is given by the determinant:
\[
W(f, g) = \begin{vmatrix}
f(x) & g(x) \\
f'(x) & g'(x)
\end{vmatrix}
\]
- Substituting \(f(x) = \sin 3x\) and \(g(x) = \cos 3x\), we find:
\[
W(\sin 3x, \cos 3x) = \begin{vmatrix}
\sin 3x & \cos 3x \\
3\cos 3x & -3\sin 3x
\end{vmatrix}
\]
- Calculate the determinant:
\[
W = (\sin 3x)(-3\sin 3x) - (\cos 3x)(3\cos 3x) = -3\sin^2 3x - 3\cos^2 3x
\]
- Simplify using the trigonometric identity \(\sin^2 a + \cos^2 a = 1\):
\[
W = -3(\sin^2 3x + \cos^2 3x) = -3 \cdot 1 = -3
\]
- Since \(W \neq 0\), the functions \(\sin 3x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb26ca370-3b85-410b-82ca-0e03b921dc6b%2F8df6de58-13dd-49ad-b87f-0feadb63ef24%2Ffwzvlmb_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
1. **Wronskian of Trigonometric Functions**
Find the Wronskian of the set \(\{\sin 3x, \cos 3x\}\) and determine if it is linearly independent or linearly dependent.
2. **Wronskian of Polynomial Functions**
Find the Wronskian of the set \(\{1 - x, 1 + x, 1 - 3x\}\).
### Explanation
The Wronskian is a determinant used in the study of differential equations to determine whether a set of functions is linearly independent.
Let's analyze both sets:
#### 1. **\(\{\sin 3x, \cos 3x\}\):**
- The Wronskian \(W(f, g)\) of two functions \(f(x)\) and \(g(x)\), in this case, \(\sin 3x\) and \(\cos 3x\), is given by the determinant:
\[
W(f, g) = \begin{vmatrix}
f(x) & g(x) \\
f'(x) & g'(x)
\end{vmatrix}
\]
- Substituting \(f(x) = \sin 3x\) and \(g(x) = \cos 3x\), we find:
\[
W(\sin 3x, \cos 3x) = \begin{vmatrix}
\sin 3x & \cos 3x \\
3\cos 3x & -3\sin 3x
\end{vmatrix}
\]
- Calculate the determinant:
\[
W = (\sin 3x)(-3\sin 3x) - (\cos 3x)(3\cos 3x) = -3\sin^2 3x - 3\cos^2 3x
\]
- Simplify using the trigonometric identity \(\sin^2 a + \cos^2 a = 1\):
\[
W = -3(\sin^2 3x + \cos^2 3x) = -3 \cdot 1 = -3
\]
- Since \(W \neq 0\), the functions \(\sin 3x
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