Applying any method and show all the steps to determine whether f,(x) = cos 2x f,(x) = 1 f,(x)= cos x are linearly independent.
Applying any method and show all the steps to determine whether f,(x) = cos 2x f,(x) = 1 f,(x)= cos x are linearly independent.
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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Question
![**Problem Statement:**
Determine whether the functions \( f_1(x) = \cos 2x \), \( f_2(x) = 1 \), and \( f_3(x) = \cos^2 x \) are linearly independent by applying any method. Show all steps clearly.
---
**Steps for Determining Linear Independence:**
A set of functions \(\{f_1, f_2, f_3\}\) is said to be linearly independent if the equation:
\[ c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \]
holds true only when \( c_1 = c_2 = c_3 = 0 \) for all \(x\).
To determine this, we need to solve:
\[ c_1 \cos 2x + c_2 \cdot 1 + c_3 \cos^2 x = 0 \]
for all possible values of \(x\).
---
**Solving the Equation:**
1. Use trigonometric identities:
- Recall the identity: \(\cos 2x = 2\cos^2 x - 1\)
2. Substitute \(\cos 2x\) in the equation:
\[
c_1 (2\cos^2 x - 1) + c_2 + c_3 \cos^2 x = 0
\]
3. Rearrange terms:
\[
(2c_1 + c_3)\cos^2 x - c_1 + c_2 = 0
\]
This equation must hold for all \(x\).
---
**Analyzing Coefficients:**
- Check if the coefficients of \(\cos^2 x\) and the constant terms can be zero independently:
1. \(2c_1 + c_3 = 0\)
2. \(-c_1 + c_2 = 0\)
- Solve the system of equations:
From equation (1):
\[
c_3 = -2c_1
\]
From equation (2):
\[
c_2 = c_1
\]
Therefore, substitute back to check for non-trivial solutions:
If \( c_1 = 0 \), then \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c1e8b90-227b-4a9c-b073-26e8337ecb0b%2F30244c9a-7634-4015-a8a6-b25e5c55da75%2Flsnidgr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine whether the functions \( f_1(x) = \cos 2x \), \( f_2(x) = 1 \), and \( f_3(x) = \cos^2 x \) are linearly independent by applying any method. Show all steps clearly.
---
**Steps for Determining Linear Independence:**
A set of functions \(\{f_1, f_2, f_3\}\) is said to be linearly independent if the equation:
\[ c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \]
holds true only when \( c_1 = c_2 = c_3 = 0 \) for all \(x\).
To determine this, we need to solve:
\[ c_1 \cos 2x + c_2 \cdot 1 + c_3 \cos^2 x = 0 \]
for all possible values of \(x\).
---
**Solving the Equation:**
1. Use trigonometric identities:
- Recall the identity: \(\cos 2x = 2\cos^2 x - 1\)
2. Substitute \(\cos 2x\) in the equation:
\[
c_1 (2\cos^2 x - 1) + c_2 + c_3 \cos^2 x = 0
\]
3. Rearrange terms:
\[
(2c_1 + c_3)\cos^2 x - c_1 + c_2 = 0
\]
This equation must hold for all \(x\).
---
**Analyzing Coefficients:**
- Check if the coefficients of \(\cos^2 x\) and the constant terms can be zero independently:
1. \(2c_1 + c_3 = 0\)
2. \(-c_1 + c_2 = 0\)
- Solve the system of equations:
From equation (1):
\[
c_3 = -2c_1
\]
From equation (2):
\[
c_2 = c_1
\]
Therefore, substitute back to check for non-trivial solutions:
If \( c_1 = 0 \), then \(
Expert Solution
Step 1
Here, use the method of Wronskian according to which Wronskian is non zero for linearly independent functions and wronskian of n functions is given by the following determinant:
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