**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 \(

Answered: Applying any method and show all the…

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 \(

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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