Chapter 4: Q27E (page 164)

In Problems 27–32, use Definition 1 to determine whether the functions y₁and y₂are linearly dependent on the interval (0, 1).
27. y₁(t) = costsint, y₂(t) = sin2t

Short Answer

Expert verified

The functions are linearly dependent y₁(t)=2Cy₂ for any C.

Step by step solution

Apply the given values.

Here y₁(t) = costsint, y₂(t) = sin2t

If the y₁ and y₂ are linearly dependent on the interval (0, 1) then

y₁(t) = C₁y₂(t)

The trigonometric identity is $\sin 2 θ = 2 \sin θ \cos θ$ .

Find linearly dependent functions.

Now,

$\begin{array}{rcl} y_{1} (t) & = & c_{1} y_{2} (t) \\ \cos (t) \sin (t) & = & 2 C_{1} \sin (t) \cos (t) \end{array}$

Therefore, y₁(t) = 2Cy₂(t), where C = 2C₁,

Thus, these functions are linear dependent.

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In Problems 27–32, use Definition 1 to determine whether the functions y₁and y₂are linearly dependent on the interval (0, 1).
27. y₁(t) = costsint, y₂(t) = sin2t

Short Answer

Step by step solution

Apply the given values.

Find linearly dependent functions.

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In Problems 27–32, use Definition 1 to determine whether the functions y1and y2are linearly dependent on the interval (0, 1).27. y1(t) = costsint, y2(t) = sin2t

Short Answer

Step by step solution

Apply the given values.

Find linearly dependent functions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Discrete Mathematics

Pure Maths

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 27–32, use Definition 1 to determine whether the functions y₁and y₂are linearly dependent on the interval (0, 1).
27. y₁(t) = costsint, y₂(t) = sin2t