Question

Diff EQ. Linearly independent? Where do they get the value x=0, to plug in and get the expression equal to 1?

Answer 1

They are using this rule:

Given two functions \(f(x)\) and \(g(x)\) that are differentiable on some interval \(I\).

(1) If \(W(f,g)(x_0)\ne0\) for some \(x_0\) in \(I\), then \(f(x)\) and \(g(x)\) are linearly independent on the interval \(I\).

(2) If \(f(x)\) and \(g(x)\) are linearly dependent on \(I\) then \(W(f,g)(x) = 0\) for all \(x\) in the interval \(I\).

They made the conclusion by choosing \(x_0=0\) and applying rule (1) above.

Answer 2

If you want to learn more there is a great site called Pauls Online Math Notes.

Here is the relevant section in that site that might help you further understand this:
http://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx

Answer 3

Thanks, I missed that.

Answer 4

yw :)