Chapter 3: Q.13P (page 136)

In Problems $8 to 15, use (8.5)$ to show that the given functions are linearly independent.
$13. \sin x, \sin 2 x$

Short Answer

Expert verified

It has been shown that the functions $\sin x, \sin 2 x$ are linearly independent except at

$x = (n + \frac{1}{2}) Π$

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

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

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 $f_{1} (x) = \sin x of order 1$

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

Find the derivatives of the function of order $f_{2} (x) = \sin 2 x of order 1$

$\begin{aligned} f_{2} (x) = \sin 2 x \\ f_{2}^{'} (x) = 2 \cos 2 x \end{aligned}$

Substitute the derivatives in the Wronksian formula and simplify as follows:

localid="1657426787997" $\begin{aligned} W = |\begin{array}{cc} \sin x & \sin 2 x \\ \cos x & 2 \cos 2 x \end{array}| \\ = 2 \sin x \cos 2 x - \sin 2 x \cos x \end{aligned}$

Use the trigonometric identities $\cos 2 x = \cos^{2} x - \sin^{2} x and \sin 2 x = 2 \sin x \cos x$

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In Problems $8 to 15, use (8.5)$ to show that the given functions are linearly independent.
$13. \sin x, \sin 2 x$

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

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In Problems8to15,use(8.5)to show that the given functions are linearly independent.13.sin⁡x,sin⁡2x

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

Quantum Physics

Classical Mechanics

Oscillations

Nuclear Physics

Work Energy and Power

Linear Momentum

Study anywhere. Anytime. Across all devices.

In Problems $8 to 15, use (8.5)$ to show that the given functions are linearly independent.
$13. \sin x, \sin 2 x$