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Differential Equations-linear dependence of homogenous equations/genEq

Problem 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

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

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