Paul's Online Math Notes
     
 
Online Notes / Differential Equations (Notes) / Second Order DE`s / More on the Wronskian
Notice
I've been notified that Lamar University will be doing some network maintenance on the following days.
  • Sunday November 9th 2014 from 12:05 AM until 11:59 AM Central Daylight Time
  • Sunday November 16th 2014 from 4:0 AM until 11:59 AM Central Daylight Time

During these times the site will either be completely unavailable or you will receive an error when trying to access any of the notes and/or problem pages. I realize this is probably a bad time for many of you but I have no control over this kind of thing and there are really no good times for this to happen and they picked the time that would cause the least disruptions for the fewest people. I apologize for the inconvenience!

Paul.



Internet Explorer 10 & 11 Users : If you are using Internet Explorer 10 or Internet Explorer 11 then, in all likelihood, the equations on the pages are all shifted downward. To fix this you need to put your browser in Compatibility View for my site. Click here for instructions on how to do that. Alternatively, you can also view the pages in Chrome or Firefox as they should display properly in the latest versions of those browsers without any additional steps on your part.

 More on the Wronskian

In the previous section we introduced the Wronskian to help us determine whether two solutions were a fundamental set of solutions.  In this section we will look at another application of the Wronskian as well as an alternate method of computing the Wronskian.

 

Let’s start with the application.  We need to introduce a couple of new concepts first.

 

Given two non-zero functions f(x) and g(x) write down the following equation.

 

(1)

 

Notice that c = 0 and k = 0 will make (1) true for all x regardless of the functions that we use.

 

Now, if we can find non-zero constants c and k for which (1) will also be true for all x then we call the two functions linearly dependent.   On the other hand if the only two constants for which (1) is true are c = 0 and k = 0 then we call the functions linearly independent.

 

Example 1  Determine if the following sets of functions are linearly dependent or linearly independent.

(a)    [Solution]

(b)    [Solution]

 

Solution

(a)   

 

We’ll start by writing down (1) for these two functions.

                                       

 

We need to determine if we can find non-zero constants c and k that will make this true for all x or if c = 0 and k = 0 are the only constants that will make this true for all x.  This is often a fairly difficult process.  The process can be simplified with a good intuition for this kind of thing, but that’s hard to come by, especially if you haven’t done many of these kinds of problems.

 

In this case the problem can be simplified by recalling

                                                   

 

Using this fact our equation becomes.

                                                   

 

With this simplification we can see that this will be zero for any pair of constants c and k that satisfy

                                                                

 

Among the possible pairs on constants that we could use are the following pairs.

                                                       

 

As I’m sure you can see there are literally thousands of possible pairs and they can be made as “simple” or as “complicated” as you want them to be.

 

So, we’ve managed to find a pair of non-zero constants that will make the equation true for all x and so the two functions are linearly dependent.

[Return to Problems]

 

(b)  

 

As with the last part, we’ll start by writing down (1) for these functions.

                                                               

 

In this case there isn’t any quick and simple formula to write one of the functions in terms of the other as we did in the first part.  So, we’re just going to have to see if we can find constants.  We’ll start by noticing that if the original equation is true, then if we differentiate everything we get a new equation that must also be true.  In other words, we’ve got the following system of two equations in two unknowns.

                                                              

 

We can solve this system for c and k and see what we get.  We’ll start by solving the second equation for c.

                                                                   

Now, plug this into the first equation.

                                                          

 

Recall that we are after constants that will make this true for all t.  The only way that this will ever be zero for all t is if k = 0!  So, if k = 0 we must also have c = 0.

 

Therefore, we’ve shown that the only way that

                                                               

will be true for all t is to require that c = 0 and k = 0.  The two functions therefore, are linearly independent.

[Return to Problems]

 

As we saw in the previous examples determining whether two functions are linearly independent or linearly dependent can be a fairly involved process.  This is where the Wronskian can help.

 

Fact

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

(1) If  for some x0 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.

 

Be very careful with this fact.  It DOES NOT say that if W(f,g)(x) = 0 then f(x) and g(x) are linearly dependent!  In fact it is possible for two linearly independent functions to have a zero Wronskian! 

 

This fact is used to quickly identify linearly independent functions and functions that are liable to be linearly dependent.

 

Example 2  Verify the fact using the functions from the previous example.

 

Solution

(a)  

 

In this case if we compute the Wronskian of the two functions we should get zero since we have already determined that these functions are linearly dependent.

                              

So, we get zero as we should have.  Notice the heavy use of trig formulas to simplify the work!

 

(b)  

 

Here we know that the two functions are linearly independent and so we should get a non-zero Wronskian.

                                                 

The Wronskian is non-zero as we expected provided .  This is not a problem.  As long as the Wronskian is not identically zero for all t we are okay.

 

 

Example 3  Determine if the following functions are linearly dependent or linearly independent.

(a)    [Solution]

(b)    [Solution]

 

Solution

(a) Now that we have the Wronskian to use here let’s first check that.  If its non-zero then we will know that the two functions are linearly independent and if its zero then we can be pretty sure that they are linearly dependent.

                                         

 

So, by the fact these two functions are linearly independent.  Much easier this time around!

[Return to Problems]

 

(b) We’ll do the same thing here as we did in the first part.  Recall that

                                                              

Now compute the Wronskian.

                                 

 

Now, this does not say that the two functions are linearly dependent!  However, we can guess that they probably are linearly dependent.  To prove that they are in fact linearly dependent we’ll need to write down (1) and see if we can find non-zero c and k that will make it true for all x.

                                                            

 

So, it looks like we could use any constants that satisfy

                                                                

to make this zero for all x.  In particular we could use

                                                       

 

We have non-zero constants that will make the equation true for all x.  Therefore, the

functions are linearly dependent.

[Return to Problems]

 

Before proceeding to the next topic in this section let’s talk a little more about linearly independent and linearly dependent functions.  Let’s start off by assuming that f(x) and g(x) are linearly dependent.  So, that means there are non-zero constants c and k so that

 

 

is true for all x.

 

Now, we can solve this in either of the following two ways.

 

 

 

Note that this can be done because we know that c and k are non-zero and hence the divisions can be done without worrying about division by zero.

 

So, this means that two linearly dependent functions can be written in such a way that one is nothing more than a constant times the other.  Go back and look at both of the sets of linearly dependent functions that we wrote down and you will see that this is true for both of them.

 

Two functions that are linearly independent can’t be written in this manner and so we can’t get from one to the other simply by multiplying by a constant.

 

Next, we don’t want to leave you with the impression that linear independence and linear dependence is only for two functions.  We can easily extend the idea to as many functions as we’d like.

 

Let’s suppose that we have n non-zero functions, f1(x), f2(x),…, fn(x).  Write down the following equation.

(2)

 

If we can find constants c1, c2, …, cn with at least two non-zero so that (2) is true for all x then we call the functions linearly dependent.  If, on the other hand, the only constants that make (2) true for x are c1 = 0,  c2 = 0, …, cn = 0 then we call the functions linearly independent.

 

Note that unlike the two function case we can have some of the constants be zero and still have the functions be linearly dependent.

 

In this case just what does it mean for the functions to be linearly dependent?  Well, let’s suppose that they are.  So, this means that we can find constants, with at least two non-zero so that (2) is true for all x.  For the sake of argument let’s suppose that c1 is one of the non-zero constants.  This means that we can do the following.

 

 

 

 

In other words, if the functions are linearly dependent then we can write at least one of them in terms of the other functions.

 

Okay, let’s move on to the other topic of this section.  There is an alternate method of computing the Wronskian.  The following theorem gives this alternate method.

 

Abel’s Theorem

If  y1(t) and y2(t) are two solutions to 

                                                      

then the Wronskian of the two solutions is

                                           

for some t0.

 

Because we don’t know the Wronskian and we don’t know t0 this won’t do us a lot of good apparently.  However, we can rewrite this as

(3)

where the original Wronskian sitting in front of the exponential is absorbed into the c and the evaluation of the integral at t0 will put a constant in the exponential that can also be brought out and absorbed into the constant c.  If you don’t recall how to do this go back and take a look at the linear, first order differential equation section as we did something similar there.

 

With this rewrite we can compute the Wronskian up to a multiplicative constant, which isn’t too bad.  Notice as well that we don’t actually need the two solutions to do this.  All we need is the coefficient of the first derivative from the differential equation (provided the coefficient of the second derivative is one of course…).

 

Let’s take a look at a quick example of this.

 

Example 4  Without solving, determine the Wronskian of two solutions to the following differential equation.

                                                         

Solution

The first thing that we need to do is divide the differential equation by the coefficient of the second derivative as that needs to be a one.  This gives us

                                                           

Now, using (3) the Wronskian is

                                               


Online Notes / Differential Equations (Notes) / Second Order DE`s / More on the Wronskian

[Contact Me] [Links] [Privacy Statement] [Site Map] [Terms of Use] [Menus by Milonic]

© 2003 - 2014 Paul Dawkins