Chapter 3: Q.15P (page 136)

In Problems 8 to 15,use to show that the given functions are linearly independent.
15. $e^{x}, e^{i x}, \cosh x$

Short Answer

Expert verified

It has been shown that the functions $e^{x}, e^{i x}, \cosh x$ are linearly independent.

Step by step solution

Definition of linearly independent functions

The functions $f_{1} (x), f_{2} (x), \dots, f_{n} (x)$ are linearly independent if the determinant

$N = |\begin{array}{cccc} f_{1} (x) & f_{2} (x) & \dots & f_{n} (x) \\ {f_{1}}^{'} (x) & {f_{2}}^{'} (x) & \dots & f_{n}^{'} (x) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ f_{1}^{(n - 1)} (x) & f_{2}^{(n - 1)} (x) & \dots & f_{n}^{(n - 1)} (x) \end{array}| \neq 0$

Here, W is called the Wronksian of functions.

Use the Wronksian to show that the given functions are linearly independent.

Find the derivatives of the function of order $f_{1} (x) = e^{x} of order n - 1 .$

$\begin{aligned} f_{1} (x) = e^{x} \\ f_{1}^{'} (x) = e^{x} \\ f_{1}^{''} (x) = e^{x} \end{aligned}$

Find the derivatives of the function of order $f_{2} (x) = e^{i x} of order n - 1$

$\begin{aligned} f_{2} (x) = e^{i x} \\ f_{2}^{''} (x) = i e^{i x} \\ f_{2}^{*} (x) = - e^{i x} \end{aligned}$

Find the derivatives of the function of order $f_{3} (x) = \cosh x of order n - 1$

$\begin{aligned} f_{3} (x) = \cosh x \\ f_{3}^{'} (x) = \sinh x \\ f_{3}^{'} (x) = \cosh x \end{aligned}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 8 to 15,use to show that the given functions are linearly independent.
15. $e^{x}, e^{i x}, \cosh x$

Short Answer

Step by step solution

Definition of linearly independent functions

Use the Wronksian to show that the given functions are linearly independent.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Turning Points in Physics

Famous Physicists

Rotational Dynamics

Electric Charge Field and Potential

Astrophysics

Study anywhere. Anytime. Across all devices.

In Problems 8 to 15,use to show that the given functions are linearly independent.15.ex,eix,cosh⁡x

Short Answer

Step by step solution

Definition of linearly independent functions

Use the Wronksian to show that the given functions are linearly independent.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Turning Points in Physics

Famous Physicists

Rotational Dynamics

Electric Charge Field and Potential

Astrophysics

Study anywhere. Anytime. Across all devices.

In Problems 8 to 15,use to show that the given functions are linearly independent.
15. $e^{x}, e^{i x}, \cosh x$