Using the Wronskian, determine if the sets are independent. a) {sin 3x, cos 3x} b) { ex, e-x}
Using the Wronskian, determine if the sets are independent. a) {sin 3x, cos 3x} b) { ex, e-x}
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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![**Topic: Using the Wronskian to Determine Linear Independence**
In this educational lesson, we will explore the concept of linear independence of sets of functions using the Wronskian determinant. This technique is pivotal in the study of differential equations and linear algebra. Below we have two sets of functions, and we will determine if they are linearly independent through the Wronskian method.
### Problem Statement
#### Using the Wronskian, determine if the sets are independent:
a) \( \{ \sin 3x, \cos 3x \} \)
b) \( \{ e^x, e^{-x} \} \)
### Explanation
The Wronskian of two functions \( f \) and \( g \), denoted as \( W(f, g) \), is calculated using the following determinant:
\[
W(f, g) = \begin{vmatrix}
f(x) & g(x) \\
f'(x) & g'(x)
\end{vmatrix}
\]
For a pair of functions to be linearly independent on an interval, their Wronskian must be non-zero at some point within that interval. Let's calculate the Wronskian for each set of functions.
#### a) \( \{ \sin 3x, \cos 3x \} \)
1. Calculate the derivatives:
\[
\frac{d}{dx}(\sin 3x) = 3\cos 3x
\]
\[
\frac{d}{dx}(\cos 3x) = -3\sin 3x
\]
2. Form the Wronskian determinant:
\[
W(\sin 3x, \cos 3x) = \begin{vmatrix}
\sin 3x & \cos 3x \\
3\cos 3x & -3\sin 3x
\end{vmatrix}
\]
3. Evaluate the determinant:
\[
W(\sin 3x, \cos 3x) = (\sin 3x)(-3\sin 3x) - (\cos 3x)(3\cos 3x)
\]
\[
W(\sin 3x, \cos 3x) = -3\sin^2 3x -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57d9078-3a26-4ab9-a5c6-f67f173c178d%2Ff3de47a5-ae9e-46b9-b4fa-f413b59656ec%2Fmk5nkdh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Using the Wronskian to Determine Linear Independence**
In this educational lesson, we will explore the concept of linear independence of sets of functions using the Wronskian determinant. This technique is pivotal in the study of differential equations and linear algebra. Below we have two sets of functions, and we will determine if they are linearly independent through the Wronskian method.
### Problem Statement
#### Using the Wronskian, determine if the sets are independent:
a) \( \{ \sin 3x, \cos 3x \} \)
b) \( \{ e^x, e^{-x} \} \)
### Explanation
The Wronskian of two functions \( f \) and \( g \), denoted as \( W(f, g) \), is calculated using the following determinant:
\[
W(f, g) = \begin{vmatrix}
f(x) & g(x) \\
f'(x) & g'(x)
\end{vmatrix}
\]
For a pair of functions to be linearly independent on an interval, their Wronskian must be non-zero at some point within that interval. Let's calculate the Wronskian for each set of functions.
#### a) \( \{ \sin 3x, \cos 3x \} \)
1. Calculate the derivatives:
\[
\frac{d}{dx}(\sin 3x) = 3\cos 3x
\]
\[
\frac{d}{dx}(\cos 3x) = -3\sin 3x
\]
2. Form the Wronskian determinant:
\[
W(\sin 3x, \cos 3x) = \begin{vmatrix}
\sin 3x & \cos 3x \\
3\cos 3x & -3\sin 3x
\end{vmatrix}
\]
3. Evaluate the determinant:
\[
W(\sin 3x, \cos 3x) = (\sin 3x)(-3\sin 3x) - (\cos 3x)(3\cos 3x)
\]
\[
W(\sin 3x, \cos 3x) = -3\sin^2 3x -
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