Chapter 4: Q26E (page 199)

Let $y_{1} (t) = t^{3}$ and $y_{2} (t) = |t^{3}|$ . Are $y_{1}$ and $y_{2}$ linearly independent on the following intervals?
(a). $[0, \infty)$
(b). $(- \infty, 0]$
(c). $(- \infty, \infty)$
(d) Compute the Wronskian $W [y_{1}, y_{2}] (t)$ on the interval $(- \infty, \infty)$ .

Short Answer

Expert verified

(a). In the interval $[0, \infty)$ , role="math" localid="1663933480488" $y_{1}$ and $y_{2}$ are linearly dependent.

(b). In the interval $(- \infty, 0]$ , $y_{1}$ and $y_{2}$ are linearly dependent.

(c). In the interval $(- \infty, \infty)$ , $y_{1}$ and $y_{2}$ are linearly independent.

(d). The solution of the interval $(- \infty, \infty)$ is $W [y_{1}, y_{2}] = 0$ .

Step by step solution

Check whether the given statement is dependent or independent

Given $y_{1} (t) = t^{3}$ and $y_{2} (t) = |t^{3}|$

Interval is $[0, \infty)$ in this interval $|t^{3}| = t^{3}$ means role="math" localid="1663933717863" $y_{1} = y_{2}$ . So $y_{1}$ and $y_{2}$ are linearly dependent.

Check whether the given statement is dependent or independent

Interval is $(- \infty, 0]$ in this interval $|t^{3}| = - t^{3}$ means $y_{1} = - y_{2}$ . So $y_{1}$ and $y_{2}$ are linearly dependent

Check whether the given statement is dependent or independent

Interval is $(- \infty, \infty)$

$c_{1} t^{3} + c_{2} |t^{3}| = 0$

The above equation is true only when $c_{1} = c_{2} = 0$ . Therefore $y_{1}$ and $y_{2}$ are linearly independent.

Compute the Wronskian

If,

$\begin{array}{rcl} y_{1} & = & t^{3} \Rightarrow y_{1}^{}' \\ = & 3 t^{2} \\ y_{2} & = & |t^{3}| \Rightarrow \frac{|t^{3}|}{t^{3}} \times 3 t^{2} \\ = & 3 \frac{|t^{3}|}{t} \end{array}$

Then,

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Let $y_{1} (t) = t^{3}$ and $y_{2} (t) = |t^{3}|$ . Are $y_{1}$ and $y_{2}$ linearly independent on the following intervals?
(a). $[0, \infty)$
(b). $(- \infty, 0]$
(c). $(- \infty, \infty)$
(d) Compute the Wronskian $W [y_{1}, y_{2}] (t)$ on the interval $(- \infty, \infty)$ .

Short Answer

Step by step solution

Check whether the given statement is dependent or independent

Check whether the given statement is dependent or independent

Check whether the given statement is dependent or independent

Compute the Wronskian

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Discrete Mathematics

Geometry

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let y1(t)=t3and y2(t)=t3. Are y1and y2linearly independent on the following intervals?(a). [0,∞)(b). (-∞,0](c).(-∞,∞)(d) Compute the WronskianWy1,y2(t)on the interval(-∞,∞).

Short Answer

Step by step solution

Check whether the given statement is dependent or independent

Check whether the given statement is dependent or independent

Check whether the given statement is dependent or independent

Compute the Wronskian

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Discrete Mathematics

Geometry

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let $y_{1} (t) = t^{3}$ and $y_{2} (t) = |t^{3}|$ . Are $y_{1}$ and $y_{2}$ linearly independent on the following intervals?
(a). $[0, \infty)$
(b). $(- \infty, 0]$
(c). $(- \infty, \infty)$
(d) Compute the Wronskian $W [y_{1}, y_{2}] (t)$ on the interval $(- \infty, \infty)$ .