Math Question: The Wronskian

[clickynext]

I'm having a bit of trouble studying for my Linear Differential Equations course and would appreciate a bit of help with this concept...

What exactly does the Wronskian of two solutions of a second order linear differential equation say? I understand that there are two significant cases for the wronskian, W = 0 and W not = 0. What do each of these cases directly mean?

Can the wronskian only be applied to an ODE of the form y'' + p(t)y' + q(t)y = 0 or does it apply to r(t)y'' p(t)y' + q(t)y = g(t) as well?

Thanks. 🙂

 

[hypn0tik]

The Wronskian essentially tells you if a set of functions are linearly dependent or not.

Edit:

The answer to your second question is 'yes'. The Wronskian is useful for computing the solutions to equations of that form using the method of variation of parameters.

Check out this Wikipedia article for more information.

 

[clickynext]

Originally posted by: hypn0tik
The Wronskian essentially tells you if a set of functions are linearly dependent or not.

Does linear independence state that the two solutions form a fundamental set? Or only that they are solutions? What else does linear independence imply here?

 

[hypn0tik]

Originally posted by: clickynext

Originally posted by: hypn0tik
The Wronskian essentially tells you if a set of functions are linearly dependent or not.

Does linear independence state that the two solutions form a fundamental set? Or only that they are solutions? What else does linear independence imply here?

See my edit above.

 

[clickynext]

Originally posted by: hypn0tik

Originally posted by: clickynext

Originally posted by: hypn0tik
The Wronskian essentially tells you if a set of functions are linearly dependent or not.

Does linear independence state that the two solutions form a fundamental set? Or only that they are solutions? What else does linear independence imply here?

See my edit above.

Oh okay, thanks.

 

[hypn0tik]

Originally posted by: clickynext

Originally posted by: hypn0tik

Originally posted by: clickynext

Originally posted by: hypn0tik
The Wronskian essentially tells you if a set of functions are linearly dependent or not.

Does linear independence state that the two solutions form a fundamental set? Or only that they are solutions? What else does linear independence imply here?

See my edit above.

Oh okay, thanks.

My bad on the late edit.

I did this stuff a couple years ago so I may be a bit rusty. Feel free to ask any other questions you may have.

I personally use Laplace Transforms to solve differential equations if I need to. I'm sure you'll learn that later on though.

 

[clickynext]

Originally posted by: hypn0tik

Originally posted by: clickynext

Originally posted by: hypn0tik

Originally posted by: clickynext

Originally posted by: hypn0tik
The Wronskian essentially tells you if a set of functions are linearly dependent or not.

Does linear independence state that the two solutions form a fundamental set? Or only that they are solutions? What else does linear independence imply here?

See my edit above.

Oh okay, thanks.

My bad on the late edit.

I did this stuff a couple years ago so I may be a bit rusty. Feel free to ask any other questions you may have.

I personally use Laplace Transforms to solve differential equations if I need to. I'm sure you'll learn that later on though.

Yeah, I'm just scraping through the basics. It's a confusing topic for me and I can't quite wrap my mind around most of the theorems in the course, unlike a more familiar topic like Calculus where I've already had some experience and basic courses on it.

Do you have any good books on it that you would recommend? One thing I noticed through taking five math courses so far is that Stewart Calculus seems to explain things in a much simpler, cleaner, more systematic way than most other books. Comparing the way the basics of DEs are explained in Stewart to the Boyce and DiPrima I'm using right now, Stewart is much easier to understand for some reason.

I appreciate the help.

 

[hypn0tik]

Originally posted by: clickynext

Yeah, I'm just scraping through the basics. It's a confusing topic for me and I can't quite wrap my mind around most of the theorems in the course, unlike a more familiar topic like Calculus where I've already had some experience and basic courses on it.

Do you have any good books on it that you would recommend? One thing I noticed through taking five math courses so far is that Stewart Calculus seems to explain things in a much simpler, cleaner, more systematic way than most other books. Comparing the way the basics of DEs are explained in Stewart to the Boyce and DiPrima I'm using right now, Stewart is much easier to understand for some reason.

I appreciate the help.

Those are precisely the books I used when I was learning this stuff. I haven't used any other books for learning DEs so I can't really tell you how the Boyce and DiPrima compares to the others. Stewart is definitely an amazing book.

I don't remember many of the theorems so I can't really help you out there. However, if you need help solving a differential equation, I may be able to help you out.

 

[krotchy]

Solving ODE's makes babies cry.

The one thing I love about my controls systems professor, is he doesnt make us solve ODE's by hand. He just wants us to understand how to model a system using ODE's and turn that into a Feedback system, and state space driagram to create an S-domain transform. Then we let matlab solve everything 🙂

 

[DrPizza]

Originally posted by: krotchy
Solving ODE's makes babies cry.

The one thing I love about my controls systems professor, is he doesnt make us solve ODE's by hand. He just wants us to understand how to model a system using ODE's and turn that into a Feedback system, and state space driagram to create an S-domain transform. Then we let matlab solve everything 🙂

I ended up taking Diff Eq's twice; it didn't transfer because the first time I took it, it concentrated on solving the equations by hand. The second time I took it, it concentrated on understanding the equations, using them to model systems, attractors, repellors, etc. Personally, I enjoyed the 2nd course far more. Now, as much as I push my calculus students to attempt really difficult problems, I usually motion toward the computer and say, "well, we each got a different answer. In the real world, that's what those things are for." I spend far more time on applications for calculus and modeling with differential equations with those students than on any other topic.

 

[Narmer]

Wronksian tells you whether or not the solutions are independent. If they are independent, then they should span the vector-space, thereby forming a vector for the basis. Wronksian is basically a Determinant.