Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q28E (page 200)

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

Short Answer

Expert verified

(a). In the interval $[0, 鈭�)$ , $y_{1}$ and $y_{2}$ are linearly dependent.

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

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

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

Step by step solution

Check whether the given statement is dependent or independent

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

Interval is role="math" localid="1664184951931" $[0, 鈭�)$ in this interval $2 t | t | = 2 t^{2}$ means $2 y_{1} = y_{2}$ . So $y_{1}$ and $y_{2}$ are linearly dependent.

Check whether the given statement is dependent or independent

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

Check whether the given statement is dependent or independent

Interval is $(- 鈭�, 鈭�)$

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

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,

$y_{1} = t^{2} 鈬� y_{1}^{}' = 2 t y_{2} = 2 t | t | 鈬� 2 | t | + 2 | t | = 4 | t |$

Then,

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91影视

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

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.

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91影视

Let y1(t)=t2andy2t=2tt. Arey1andy2linearly independent on the interval(a). [0,鈭�)(b). (-鈭�,0](c). (-鈭�,鈭�)(d). Compute the Wronskian Wy1,y2(t)on the interval (-鈭�,鈭�).

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

Discrete Mathematics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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