Applying any method and show all the steps to determine whether f,(x) = cos 2x f,(x) =1 f,(x) = cosx are linearly independent.
Applying any method and show all the steps to determine whether f,(x) = cos 2x f,(x) =1 f,(x) = cosx 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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![**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 and showing all steps.
**Explanation:**
To determine the linear independence of the given functions, start by checking if there exist constants \( c_1, c_2, \) and \( c_3 \), not all zero, such that:
\[ c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \]
for all \( x \).
**Steps:**
1. **Substitute the given functions:**
\[ c_1 \cos 2x + c_2 \cdot 1 + c_3 \cos^2 x = 0 \]
2. **Use trigonometric identities:**
Recall the identity: \( \cos 2x = 2\cos^2 x - 1 \).
Substitute back:
\[ c_1 (2\cos^2 x - 1) + c_2 + c_3 \cos^2 x = 0 \]
3. **Simplify the equation:**
\[ (2c_1 + c_3) \cos^2 x - c_1 + c_2 = 0 \]
Set the coefficients of like terms equal to zero:
\[
\begin{align*}
2c_1 + c_3 &= 0 \\
-c_1 + c_2 &= 0
\end{align*}
\]
4. **Solve the system of equations:**
From \( 2c_1 + c_3 = 0 \), we get \( c_3 = -2c_1 \).
From \(-c_1 + c_2 = 0\), we get \( c_2 = c_1 \).
Thus the solution is of the form \( c_1(1, 1, -2) \).
5. **Conclusion:**
Since the constants \( c_1, c_2, \) and \( c_3 \) are not all zero, the functions \( f_1(x), f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c1e8b90-227b-4a9c-b073-26e8337ecb0b%2F3c7affec-bbc6-418a-9282-0fea88a20db8%2Fj8rtty_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 and showing all steps.
**Explanation:**
To determine the linear independence of the given functions, start by checking if there exist constants \( c_1, c_2, \) and \( c_3 \), not all zero, such that:
\[ c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \]
for all \( x \).
**Steps:**
1. **Substitute the given functions:**
\[ c_1 \cos 2x + c_2 \cdot 1 + c_3 \cos^2 x = 0 \]
2. **Use trigonometric identities:**
Recall the identity: \( \cos 2x = 2\cos^2 x - 1 \).
Substitute back:
\[ c_1 (2\cos^2 x - 1) + c_2 + c_3 \cos^2 x = 0 \]
3. **Simplify the equation:**
\[ (2c_1 + c_3) \cos^2 x - c_1 + c_2 = 0 \]
Set the coefficients of like terms equal to zero:
\[
\begin{align*}
2c_1 + c_3 &= 0 \\
-c_1 + c_2 &= 0
\end{align*}
\]
4. **Solve the system of equations:**
From \( 2c_1 + c_3 = 0 \), we get \( c_3 = -2c_1 \).
From \(-c_1 + c_2 = 0\), we get \( c_2 = c_1 \).
Thus the solution is of the form \( c_1(1, 1, -2) \).
5. **Conclusion:**
Since the constants \( c_1, c_2, \) and \( c_3 \) are not all zero, the functions \( f_1(x), f
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