- #1

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**Let y1(t) = t^2 and y2(t) = t|t|**

(a) Show that y1 and y2 are linearly dependent on the interval 0<=t<=1.

(b) Show that y1 and y2 are linearly independent on the interval -1<=t<=1.

(c) Show that the Wronskian W[y1,y2] is identically zero.

(a) Show that y1 and y2 are linearly dependent on the interval 0<=t<=1.

(b) Show that y1 and y2 are linearly independent on the interval -1<=t<=1.

(c) Show that the Wronskian W[y1,y2] is identically zero.

**My attempt:**

(a) So I know that if the Wronskian is zero, then y1 and y2 are linearly dependent. So i worked out the wronskian and i got

w[y1,y2](t) = t^3 - t^2|t|

I'm not sure how to show that it is linearly dependent on the interval. should i substitute in the intervals 0 and 1 and find that they are equal to zero? what about everything inbetween 0 and 1?

(b) So for linearly independence, wronskian doesn't equal zero, again i have the same question of whether i substitute in the interval, and what do i do with the numbers inbetween, i am unsure how to prove this.

(c) i have no idea what identically zero means, can someone explain this?

(a) So I know that if the Wronskian is zero, then y1 and y2 are linearly dependent. So i worked out the wronskian and i got

w[y1,y2](t) = t^3 - t^2|t|

I'm not sure how to show that it is linearly dependent on the interval. should i substitute in the intervals 0 and 1 and find that they are equal to zero? what about everything inbetween 0 and 1?

(b) So for linearly independence, wronskian doesn't equal zero, again i have the same question of whether i substitute in the interval, and what do i do with the numbers inbetween, i am unsure how to prove this.

(c) i have no idea what identically zero means, can someone explain this?