Chapter 4: Q30E (page 164)

In Problems 27–32, use Definition 1 to determine whether the functions y₁and y₂are linearly dependent on the interval (0, 1).
30. y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)

Short Answer

Expert verified

The functions are linearly independent.

Step by step solution

Apply the given values.

The given functions are y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)

If the y₁ and y₂are linearly dependent on the interval (0, 1) then

y₁(t) = C₁y₂(t)

Find linearly dependent functions.

Check whether the function can be written as the constant multiple of others.

$\begin{array}{rcl} y_{1} (t) & = & C y_{2} (t) \\ t^{2} \cos (\ln t) & = & C t^{2} \sin (\ln t) \\ C & = & c o t (\ln t) \end{array}$

Sinceis cot(lnt) a non-constant function of t.

Therefore, these functions are linear independent.

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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).
30. y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)

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).30. y1(t) = t2cos(lnt), y2(t) = t2sin(lnt)

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

Pure Maths

Probability and Statistics

Discrete Mathematics

Statistics

Mechanics Maths

Geometry

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).
30. y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)