• Support PF! Buy your school textbooks, materials and every day products via PF Here!

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:

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.


Science Advisor
Homework Helper
Insights Author
Gold Member
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.

Want to reply to this thread?

"Differential Equations-linear dependence of homogenous equations/genEq" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving