D.E. Wronskian Method: Stuck trying to show L.I. and L.D. intervals

[Jeff12341234]

I need to show the intervals where the wronskian is linearly independent and linearly dependent.. I don't know how to do that.. Here's what I have:

 

[Dick]

Odd question. The functions are linearly independent on an interval if the Wronskian is nonzero ANYWHERE in the interval.

 

[Jeff12341234]

So what is the next procedural step?

 

[Dick]

So what is the next procedural step?

Procedure is done. Next step is to think about it. The wronskian vanishes only at 0 and 2/7. Can you think of any intervals where the wronskian vanishes? Don't think too hard. I don't think the question makes much sense.

 

[LCKurtz]

Did you understand Dick's reply? There is no positive length interval on which those functions are linearly dependent. Once you see that, there's nothing left to do. Unless you want to consider a single point as an interval.

[Edit] I didn't see Dick's second reply, sorry.

 

[Jeff12341234]

No, I didn't understand it. I need to see some sort of visual, physical representation of what's going on. :/ It's all just made up vocabulary for made up concepts at this point. All I know is that if you can solve for x, you call it the vocabulary word, Linearly dependent. If not, you call it linearly independent.

 

[Dick]

Did you understand Dick's reply? There is no positive length interval on which those functions are linearly dependent. Once you see that, there's nothing left to do. Unless you want to consider a single point as an interval.

[Edit] I didn't see Dick's second reply, sorry.

That's ok. An interval consisting of a single point is also pretty uninteresting. That makes the two functions just numbers. Always linearly dependent on single point intervals no matter what the wronskian. I can't figure out whether the person that wrote the question doesn't understand wronskians or is just trying to make a 'trick' question.

 

[LCKurtz]

That's ok. An interval consisting of a single point is also pretty uninteresting. That makes the two functions just numbers. Always linearly dependent on single point intervals no matter what the wronskian. I can't figure out whether the person that wrote the question doesn't understand wronskians or is just trying to make a 'trick' question.

I was wondering the same thing. This sort of problem usually comes up where the point is to show the student how the Wronskian vs. linear dependence differs for two solutions of a 2nd order linear DE vs. two arbitrary functions. But this exercise completely misses the target if indeed that was the target.

 

[Dick]

No, I didn't understand it. I need to see some sort of visual, physical representation of what's going on. :/ It's all just made up vocabulary for made up concepts at this point. All I know is that if you can solve for x, you call it the vocabulary word, Linearly dependent. If not, you call it linearly independent.

Two functions are said to be linearly dependent on an interval if there are two constants a and b not both 0 such that a*f1(x)+b*f2(x)=0 on the whole interval. If not then they are linearly independent. That's all. What's important here is that you learn what the wronskian has to do with that. I told you but also see here. http://en.wikipedia.org/wiki/Wronskian#The_Wronskian_and_linear_independence

 

[Jeff12341234]

So what should I write to answer part a and part b?

 

[Dick]

So what should I write to answer part a and part b?

Given what you've been told, tell us what you think. Take the interval (1,2). Linearly dependent or independent? Why? How about (-100,100)?

 

[Jeff12341234]

In my mind, there's no connection for either of the phrases, "linearly independent/dependent" to actual concepts. "Two functions are said to be linearly dependent on an interval if there are two constants a and b not both 0 such that a*f1(x)+b*f2(x)=0 on the whole interval." doesn't mean anything to me either. What interval? What constants?

What would help is a heavily generalized example/explanation that goes like this, "Some guy was doing ____ and he needed to know _____ so the wronskian test was invented to tell him _____ aspect about _____."

 

[Dick]

In my mind, there's no connection for either of the phrases, "linearly independent/dependent" to actual concepts. "Two functions are said to be linearly dependent on an interval if there are two constants a and b not both 0 such that a*f1(x)+b*f2(x)=0 on the whole interval." doesn't mean anything to me either. What interval? What constants?

What would help is a heavily generalized example/explanation that goes like this, "Some guy was doing ____ and he needed to know _____ so the wronskian test was invented to tell him _____ aspect about _____."

"Some guy was solving a third order differential equation and got three solutions, say f1(x), f2(x) and f3(x). This guy knows that if the three solutions are linearly independent then ANY solution of the differential equation can be written as a*f1(x)+b*f2(x)+c*f3(x) for arbitrary constants a,b and c, and he has a general solution. If not, then he doesn't have the complete solution. So he checks the wronskian. If there is a point where the wronskian is nonzero, then he's all done. If it's identically zero, then the guy should suspect he may have missed a solution."

 

[Jeff12341234]

"Some guy was solving a third order differential equation and got three solutions, say f1(x), f2(x) and f3(x). This guy knows that if the three solutions are linearly independent then ANY solution of the differential equation can be written as a*f1(x)+b*f2(x)+c*f3(x) for arbitrary constants a,b and c, and he has a general solution. If not, then he doesn't have the complete solution. So he checks the wronskian. If there is a point where the wronskian is nonzero, then he's all done. If it's identically zero, then the guy should suspect he may have missed a solution."

1. What's another way to say "linearly independent"? What are synonyms of the phrase? If you were making up a phrase to describe whatever linearly independent describes, what are some alternate names you would come up with for it?
2. "If there is a point where the wronskian is nonzero" what does that mean? Are you trying to say, after you set the wronskian equal to zero and solve it for x, if there are values for x the wronskian is nonzero?
3. "If it's identically zero" What does that mean?

 

[Dick]

1. What's another way to say "linearly independent"? What are synonyms of the phrase? If you were making up a phrase to describe whatever linearly independent describes, what are some alternate names you would come up with for it?
2. "If there is a point where the wronskian is nonzero" what does that mean? Are you trying to say, after you set the wronskian equal to zero and solve it for x, if there are values for x the wronskian is nonzero?
3. "If it's identically zero" What does that mean?

1. f1=x, f2=x^2 and f3=3x-2x^2 are linearly dependent, because f3=3*f1-2*f2. One of the functions can be expressed in terms of the others in a linear way. I think 'linearly dependent' sums it up pretty well.

2. Your wronskian is nonzero at x=1, x=2, x=3, x=pi, x=(-1), x=0.3182, etc etc. The only places it's zero are x=0 and x=2/7. You solved it. I'm not sure what you are asking.

3. Work out the wronskian of x, x^2 and 3x-2x^2 if you want to see what "identically zero" means.

 

[Jeff12341234]

1. So it would seem that a more intuitive definition of 'linearly dependent' would be that "one of the functions can be made using the other functions" or, "the other functions can be substituted into the last function" or "the last function can be comprised of the other functions" or something to that effect followed by an example. That tells me a LOT more than the 'standard' definition does.

2. so it's nonzero everywhere except for 0 and 2/7? If so, that's simple enough. What does that have to do with anything though? What does that actually tell me?

3. When you work that out, you just get 0

So to summarize, when solving a D.E., 2nd order or higher, after you get your answer, you use the Wronskian test to see if it's right? For the Wronskian test, you set the result equal to zero and solve for x. If you can find any values that solve for x then it gets labeled "linearly dependent" and that tells you what... that the answer is correct?? If you can't solve for x then it gets labeled as "linearly independent". (I know what linearly independent and dependent mean now, but I don't know how/why it's relevant to anything.) Finally, if you get 0 right off the bat when doing the wronskian, you messed up somewhere when solving the D.E.

 

[Dick]

2. so it's nonzero everywhere except for 0 and 2/7? If so, that's simple enough. What does that have to do with anything though? What does that actually tell me?

It tells you the functions are linearly independent. You don't even have to solve for x to know your wronskian is nonzero. Just put in x=1. You get -20e^(-3)(5). That's nonzero. So the functions are linearly independent on any interval containing x=1.

 

[Jeff12341234]

But I wrote down that they are linearly dependent which I based off of my notes. I don't know what to put for part A and part B. (2/7, inf)? (-inf, 0)? (0, 2/7)?

This supposed to be a test to see if your answer is right but getting any kind of meaning out of the results of the test is where I don't follow.. So you solve for x and you either get one or more x values or you don't. Where's the connection between that, and knowing if your general solution to the D.E. you were trying to solve is correct? If you get results for when solving for x, that means the solutions are linearly dependent. "If there is a point where the wronskian is nonzero, then he's all done" It's nonzero everywhere except 0 and 2/7 so that means the answer is right... I guess, but that doesn't tell me what to put for part A and B

 

[Dick]

Start with your guess of (2/7,inf) as an answer for A. Are the functions linearly independent on (2/7,inf)? Why?

 

[Jeff12341234]

I don't have any idea how to answer your question. I have no idea how to tell if the functions are L.I. or L.D. on a certain interval. Just guessing, I plugged in 2/7 into y1 and y2. That gave me 5e^(4/7) and 16/49*e^(-10/7) ...Is that supposed to mean something?

 

[Dick]

I don't have any idea how to answer your question. I have no idea how to tell if the functions are L.I. or L.D. on a certain interval. Just guessing, I plugged in 2/7 into y1 and y2. That gave me 5e^(4/7) and 16/49*e^(-10/7) ...Is that supposed to mean something?

Why do you think that has anything to do with linear independence? The wronskian tells you something about linear independence.

 

[Jeff12341234]

Why do you think that has anything to do with linear independence? The wronskian tells you something about linear independence.

Yea, but I don't understand how intervals come into play. I just take the functions (y1, y2, y3, etc), make them into a square matrix, run the determinant function on the square matrix, set answer equal to zero, solve for x, if x can be solved for the label "Linearly Dependent" gets written down, other wise, it's labeled L.I. I also know that for L.D. functions, the last one can be recreated using the previous functions. How that applies to anything or can be used for anything i don't know. That's all I understand so far. None of that helps me answer the question.

 

[Dick]

Yea, but I don't understand how intervals come into play. I just take the functions (y1, y2, y3, etc), make them into a square matrix, run the determinant function on the square matrix, set answer equal to zero, solve for x, if x can be solved for the label "Linearly Dependent" gets written down, other wise, it's labeled L.I. I also know that for L.D. functions, the last one can be recreated using the previous functions. How that applies to anything or can be used for anything i don't know. That's all I understand so far. None of that helps me answer the question.

If the wronskian is nonzero at any point on the interval then the functions are linearly independent on the interval. Period. That's all you need to know. Don't bring a bunch of other confusion in that will interfere with that.

 

[Ray Vickson]

Yea, but I don't understand how intervals come into play. I just take the functions (y1, y2, y3, etc), make them into a square matrix, run the determinant function on the square matrix, set answer equal to zero, solve for x, if x can be solved for the label "Linearly Dependent" gets written down, other wise, it's labeled L.I. I also know that for L.D. functions, the last one can be recreated using the previous functions. How that applies to anything or can be used for anything i don't know. That's all I understand so far. None of that helps me answer the question.

If the Wronskian is ≠ 0 at any value of x at all, then the functions are linearly independent on ANY interval whatsoever. Pick any interval you like; the functions are linearly independent on that interval. Pick another interval. The functions are also linearly independent on that interval. What, exactly, do you not understand about those statements?

 

[Jeff12341234]

ok, so this set of solutions is L.I. along (2/7, inf), (-inf, 0), and (0, 2/7) and L.D. on the point x=0 and x=2/7 ?

 

[Dick]

ok, so this set of solutions is L.I. along (2/7, inf), (-inf, 0), and (0, 2/7) and L.D. on the point x=0 and x=2/7 ?

Are they linearly independent on (-inf,inf)? Don't worry about x=0 and x=2/7, yet.

 

[Jeff12341234]

According to your previous statement they are. ("If the Wronskian is ≠ 0 at any value of x at all, then the functions are linearly independent on ANY interval whatsoever.") The wronskian equals zero at 2 points so everywhere else it doesn't equal zero. Therefore, since it qualifies for the "doesn't equal zero at any point at all" condition, they must be L.I. That logic would also mean the the wronskian is going to be L.I. like 99.99% of the time unless you get 0 as an answer.

 

[Dick]

According to your previous statement they are. ("If the Wronskian is ≠ 0 at any value of x at all, then the functions are linearly independent on ANY interval whatsoever.") The wronskian equals zero at 2 points so everywhere else it doesn't equal zero. Therefore, since it qualifies for the "doesn't equal zero at any point at all" condition, they must be L.I.

That wasn't my statement, that was Ray Vickson's statement and I wouldn't agree with it. Mine was, "If the wronskian is nonzero at any point on the interval then the functions are linearly independent on the interval." To justify that they are linearly independent in (-inf,inf) just give a value of x in (-inf,inf) where the wronskian is nonzero.

 

[Jeff12341234]

ok. It's nonzero everywhere except x=0 and x=2/7.. soo... it's L.I. ? If no interval is given by the question, I guess (-inf,inf) is assumed to be the interval.

 

[Dick]

ok. It's nonzero everywhere except x=0 and x=2/7.. soo... it's L.I. ? If no interval is given by the question, I guess (-inf,inf) is assumed to be the interval.

I picked (-inf,inf) because part A is asking for the largest interval where they are LI. (-inf,inf) is the largest interval I can think of. And yes, they are LI.

 

[Jeff12341234]

w = wronskian (determinant result)

If w = 0
--function set is L.D.
else
--function set is L.I. everywhere

If w = some function
--set w = 0
--solve for x
--If you can solve for x
----function set is L.I. except at x value(s)
--else
----function set is L.I. everywhere

Is that right?

source: "Here W=0 only when x=0 . Therefore x^2 and x are independent except at x=0 . " http://planetmath.org/WronskianDeterminant.html

 

[LCKurtz]

w = wronskian (determinant result)

If w = 0
--function set is L.D.

That is false in general. If the Wronskian of two functions is nonzero anywhere on an interval, the functions are linearly independent on that interval. But the Wronskian of two functions can be identically zero on an interval yet the functions are linearly independent on that interval. It makes a difference if the functions are solutions of a second order linear differential equation with certain hypotheses. Only then can you say that their Wronskian is either identically zero or never zero. Surely your text has a theorem about this.

 

[Dick]

w = wronskian (determinant result)

If w = 0
--function set is L.D.
else
--function set is L.I. everywhere

If w = some function
--set w = 0
--solve for x
--If you can solve for x
----function set is L.I. except at x value(s)
--else
----function set is L.I. everywhere

Is that right?

source: "Here W=0 only when x=0 . Therefore x^2 and x are independent except at x=0 . " http://planetmath.org/WronskianDeterminant.html

Mostly correct. If you read the wikipedia article carefully, it will give you an example of two functions where w=0, but they are linearly independent. w=0 usually means they are linearly dependent, but there are exceptions. To show functions are linearly independent on an interval (a,b) you don't HAVE to solve for x. All you have to do is find a point x in (a,b) where w(x)≠0. If solving for x is the way you want to do it, that's fine.

Planetmath is usually a good source, but I'm going to have to take issue with their saying x, x^2 are LI except at 0. There are LI on (-inf,inf). No 0 exceptions. Remember all you need is ONE POINT in an interval where w(x)≠0 to show LI. The fact there are isolated points where w=0 doesn't change anything.

If you want to talk about linear independence on an interval consisting of a single point, like [0,0] then that is a whole different issue and doesn't have anything to do with wronskians. On a single point interval both functions are just numbers, and two numbers are always linearly dependent, whether the wronskian is zero or not.

 

[Jeff12341234]

I looked back over that wiki article. I can't make sense of it enough to modify the "procedural code" that I previously posted.

 

[Dick]

I looked back over that wiki article. I can't make sense of it enough to modify the "procedural code" that I previously posted.

Beats me. I don't know what else to say that I haven't already repeated several times.

 

[Jeff12341234]

I need an actual procedure to perform to test for those rare exceptions you mentioned for instances where w=0 and the functions are still L.I.

 

[Dick]

I need an actual procedure to perform to test for those rare exceptions you mentioned for instances where w=0 and the functions are still L.I.

Take the wikipedia example of f1(x)=x|x| and f2(x)=x^2. Go back to the definition of linear independence. It says try to find constants a and b, not both 0 such that a*f1(x)+b*f2(x)=0 for all x. Put in x=1, that gives a+b=0. Put in x=(-1), that gives -a+b=0. Solve those two equations for a and b and conclude the only possibility is a=b=0. So they are linearly independent.

 

[Jeff12341234]

so basically the wronskian fails in these circumstances since performing the test results in w=0 implying L.D.

Another procedural test needs to be used. Could that test be to solve a system of equations such that the first equation, a*f1(x)+b*f2(x)=0, x=1, and the second equation,a*f1(x)+b*f2(x)=0, x=-1. If a=b=0 the set of functions are L.I. Can I use that procedure and be confident that I've covered all of the possibilities now? If the test I just described can always be used, why not use it in place of the wronskian?

 

[Dick]

so basically the wronskian fails in these circumstances since performing the test results in w=0 implying L.D.

Another procedural test needs to be used. Could that test be to solve a system of equations such that the first equation, a*f1(x)+b*f2(x)=0, x=1, and the second equation,a*f1(x)+b*f2(x)=0, x=-1. If a=b=0 the set of functions are L.I. Can I use that procedure and be confident that I've covered all of the possibilities now? If the test I just described can always be used, why not use it in place of the wronskian?

You can't always pick x=-1 and x=1. If you pick the two functions x^2 and x^4 then that procedure will give you a+b=0 and a+b=0 for both equations. But the wronskian will tell you they are linearly independent. Better there to pick x=1 and x=2. So it's not really a fixed procedure, you have to think about the specific functions and make sure that you choose good points. If you want to show functions are linearly independent that way, sure, you can use it instead of the wronskian.

 

[Jeff12341234]

ok. That's good to know.

Going back to the original question of this thread, I just ended up writing that the functions are L.I. everywhere except for x=0 and x=2/7 at which, on those points, they are L.D. Is there a better way to answer part A and and B that what I wrote?

 

[Dick]

ok. That's good to know.

Going back to the original question of this thread, I just ended up writing that the functions are L.I. everywhere except for x=0 and x=2/7 at which, on those points, they are L.D. Is there a better way to answer part A and and B that what I wrote?

I already said I disagree with the Planetmath phrasing in post #33. But if want to ignore that and you like it, fine. But the question is asking for intervals, isn't it?

 

[Jeff12341234]

Well, you said that pseudo code I wrote earlier was right except for special cases when w=0, which we went over. So I'm guessing the pseudo code should really be:

--If you can solve for x
----function set is L.I. everywhere

instead of:

--If you can solve for x
----function set is L.I. except at x value(s)

 

[Dick]

Well, you said that pseudo code I wrote earlier was right except for special cases when w=0, which we went over. So I'm guessing the pseudo code should really be:

--If you can solve for x
----function set is L.I. everywhere

instead of:

--If you can solve for x
----function set is L.I. except at x value(s)

If you solve for x and you only get a finite number of solutions, then the wronskian is still has a nonzero value on any interval whose length isn't zero. So, yes, they are linearly independent on any interval whose length isn't equal to zero.

 

[Jeff12341234]

So for this question, when asked where the function set is L.I., (-inf, inf) would be the best answer and for L.D., it would be 'none'?

 

[Dick]

So for this question, when asked where the function set is L.I., (-inf, inf) would be the best answer and for L.D., it would be 'none'?

That is exactly what I would say. Though for B you might note if the interval contains only a single point then they are LD, for reasons that have nothing to do with wronskians. So I wouldn't consider those exceptions. I as said way back, that part of the question is kind of goofy.

 

[Jeff12341234]

I did some more research on this. Appearently you're allowed to plug in an x value first (typically x=0) to the square matrix BEFORE doing the determinant. After that the logic is simply:

If w = 0
--function set is L.D. and is NOT the whole solution
else
--function set is L.I. and is the general solution

When I do that for this problem, i get a matrix where the top row is 5 , 0 and the second row is 10 , 0. This results in a determinant that is 0... making it L.D. I really need a once-and-for-all, authoritative proof for what this should be because I'm getting all kinds of conflicting results. To top it all off, he's asking for intervals for each which isn't even mentioned in most resources I've read.

 

[Dick]

I did some more research on this. Appearently you're allowed to plug in an x value first (typically x=0) to the square matrix BEFORE doing the determinant. After that the logic is simply:

If w = 0
--function set is L.D. and is NOT the whole solution
else
--function set is L.I. and is the general solution

When I do that for this problem, i get a matrix where the top row is 5 , 0 and the second row is 10 , 0. This results in a determinant that is 0... making it L.D. I really need a once-and-for-all, authoritative proof for what this should be because I'm getting all kinds of conflicting results. To top it all off, he's asking for intervals for each which isn't even mentioned in most resources I've read.

That is incorrect. The wronskian is 0 at x=0. You knew that from the beginning, right? It's also zero at 2/7. It doesn't change the fact that the functions are linearly independent. Here's a definitive statement. f1 and f2 are linearly dependent on an interval if and only if there are constants not both zero such that a*f1(x)+b*f2(x)=0 on the interval. That's 'definitive' because that is the 'definition' of linear dependence. Here's another definitive statement. The wronskian is zero on an interval if the functions are linearly dependent. I know that because I know how to prove it. That means if the wronskian is nonzero anywhere on an interval then they are not linearly dependent. Hence they are linearly independent. That's definitive because it's just logic.

Here's another definitive statement. I am getting really sick of the sound of my own voice as I repeat these things over and over again. If your research is turning up conflicting things then you will have to decide what is correct. I've told you everything I know about the subject several times. You decide.

 

[LCKurtz]

And for my parting shot I will just observe that the two original functions can't be solutions of a second order linear DE with nonzero leading coefficient in the first place because their Wronskian is neither nonzero nor identically zero.

And Dick, you deserve a medal for your persistence. But wait..., you already have one.

 

[Jeff12341234]

For part A) L.I. on the intervals (-inf, 0) U (0, 2/7) U (2/7). Largest interval is (-inf, 0)
..-------B) L.D. at x=0 and x=2/7

 

[Dick]

And for my parting shot I will just observe that the two original functions can't be solutions of a second order linear DE with nonzero leading coefficient in the first place because their Wronskian is neither nonzero nor identically zero.

And Dick, you deserve a medal for your persistence. But wait..., you already have one.

Thanks, LCKurtz. I decided to actually try to practice patience on this one, I thought it might work. Not sure it did. And sure, good point. I agree totally. Solutions to nice ODE's will allow you to say wronskian=0 means linearly dependent.