Differential Equations-linear dependence of homogenous equations/genEq

[cookiemnstr510510]

Homework Statement

Use definition 1 to determine whether functions Y1 and Y2 are linearly dependent on the interval (0,1). Y1=e^(3t), Y2=e^(-4t)

Relevant Equations

Definiton 1: We say Y1 and Y2 are linearly dependent on I if one of them is a constant multiple of the other on all of I

Here is my attempt at the solution:
Y1(t)=kY2(t)→e^(3t)=ke^(-4t)→(e^(3t))/(e^(-4t))=k→e^(7t)=k

So I have found a constant multiple of Y2(t), its the whole "interval" part that I don't get.
The interval is (0,1), I guess I don't really know what they are trying to say...are they saying from 0 to 1 on the x-axis this thing has to be defined? if so then e^7(0)=1 which is on that interval, but e^7(1)=a very large number which is not on that interval?

Any insight would be appreciated.
Thanks

 

[LCKurtz]

You have shown that the ratio of the two functions is $e^{7t}$, which is not constant since it depends on $t$. So one is not a constant multiple of the other and they are not linearly dependent.

 

[cookiemnstr510510]

Ahhhhh, okay. that makes a lot of sense !
Thank you.