Let y1(t)=t2 and y2t=2tt. Are y1 and y2 linearly independent on the interval (a). [0,∞)(b). (-∞,0](c). (-∞,∞)(d). Compute the Wronskian Wy1,y2(t) on the interval (-∞,∞).

Q28E - Fundamentals Of Differential Equations And Boundary Value Problems

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). $[0, \infty)$

(b). $(- \infty, 0]$

(c). $(- \infty, \infty)$

(d). Compute the Wronskian $W [y_{1}, y_{2}] (t)$ on the interval $(- \infty, \infty)$ .

Verified

Answer

(a). In the interval $[0, \infty)$ , $y_{1}$ 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 $W [y_{1}, y_{2}] = 0$ .

1Step 1: Check whether the given statement is dependent or independent

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

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

2Step 2: Check whether the given statement is dependent or independent

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

3Step 3: Check whether the given statement is dependent or independent

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

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

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

4Step 4: Compute the Wronskian

If,

$y_{1} = t^{2} \Rightarrow y_{1}^{}' = 2 t y_{2} = 2 t | t | \Rightarrow 2 | t | + 2 | t | = 4 | t |$

Then,