Question

anyone know how to solve linearly independent solutions?

Answer 1

yes

Answer 2

how do I use the Wronskian method to show that f1(x) = sinh(ax) and f2(x) = cosh(ax) are linearly independent?

Answer 3

Oh, you use the definition of the Wronskian and if the determinant you get is non-zero, the solutions are linearly independent.

Answer 4

The first row will be f1, f2, and the second row will have f1' and f2'. Then the determinant is f1f2'-f2f1'

Answer 5

and it also includes a differential equation of y'' - a^2y = 0, I have to show that they are linearly independent for that equation

Answer 6

does that make sense?

Answer 7

\[\sinh(ax)(\cosh(ax))'-\cosh(ax)(\sinh(ax))'=\sinh^2(ax)-\cosh^2(ax)=-1\]

Answer 8

a>0

Answer 9

Yeah, you can use the Wronskian to show that two solutions will be linearly independent. You would have obtained those solutions from the differential equation. Then you plug them into det(W) to see if they're independent.

Answer 10

The above expression I wrote out (that go cut off) equals -1, which is non-zero. Your solutions are linearly independent.

Answer 11

is this the only method, or easiest method, so solving these kind of questions? L.I equations?

Answer 12

to*

Answer 13

Just one sec...distracted.

Answer 14

Are you sure the 'a' in the equation isn't a^2?

Answer 15

sorry - i just checked...off sick today

Answer 16

Haha uhh yes, its a^2

Answer 17

Whoops

Answer 18

Re. your question about whether this is the easiest method to solve these equations: how are you exactly coming up with your solutions?

Answer 19

Or are you asking if this is the easiest way to show linear independence?

Answer 20

If it's the latter, I would say, 'yes'.

Answer 21

yes, I meant the latter

Answer 22

And okay, thanks! I am just learning this stuff, clearly. Newb

Answer 23

No probs. Increase my fan base - become a fan!

Answer 24

Sure thing, are you a professor? Incredibly smart student?

Answer 25

Little from column A, little from column B - postgrad.

Answer 26

Cool. Well I have no idea how to do anything on here, how do I become your fan?

Answer 27

And, why does everything I type show up five hundred times?

Answer 28

There should be a blue link next to my name that says, "Become a fan".

Answer 29

Next to a blue 'thumbs up' icon.

Answer 30

Hmm... nope

Answer 31

Try refreshing the page...I hear that works sometimes.

Answer 32

Ah well, I'll find it sometime, gotta get back to studying. Thanks for the help! I will become your fan asap. :) Appreciate it.

Answer 33

Ah! Found it

Answer 34

awesome

Answer 35

there. have a good one!

Answer 36

you too

Answer 37

nice