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

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

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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.

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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

Famous Physicists

Space Physics

Circular Motion and Gravitation

Turning Points in Physics

Force

Medical Physics

Study anywhere. Anytime. Across all devices.

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