Differential Equations: Wronskian question.

[tarmon.gaidon]

Homework Statement

Hey Everyone,

Here is a problem from my book that has my confused. I really don't understand what it wants me to do so if anyone could give me a few hints it would be greatly appreciated.

I am doing problem 34, but I included 33 since it wanted to follow the same method.

Sorry if I seem like I am asking you to do my homework. I'm not, just looking for a place to start.

Thanks,
Rob

 

[Mark44]

the attachment is invalid...

 

[tarmon.gaidon]

the attachment is invalid...

Really? I am able to open it just fine. Can anyone else open this?

EDIT: I take that back. It works for me in Chrome but not Firefox. I will upload it somewhere else and fix my post.

 

[Office_Shredder]

Let's start by looking at 33. Do you know how to use the Wronskian to show that functions are linearly independent in general?

 

[tarmon.gaidon]

Hey office_Shredder,

I am reasonably familiar with using the Wronskian to show that functions are linearly independent. I am more used to this form:

Where if W = 0 is true on an open interval I then the functions are linearly independent.

I don't completely understand the wronskian equation given in problem 33.

 

[gabbagabbahey]

Hey office_Shredder,

I am reasonably familiar with using the Wronskian to show that functions are linearly independent. I am more used to this form:

Where if W = 0 is true on an open interval I then the functions are linearly independent.

I don't completely understand the wronskian equation given in problem 33.

If you define $f_i(x)=\text{exp}(r_i x)$, $1\leq i \leq n$, what is $f'_i(x)$? How about $f''_i(x)$? What does that make the Wronskian for the $n=3$ case? Is there a rule for taking determinants where a column is multiplied by some factor that might help you here?

 

[tarmon.gaidon]

I understand where the matrix comes from now but I am not sure what method you are talking about for solving the determinant. Care to shed some light?

 

[gabbagabbahey]

Use property 3 here. For example,

$$\begin{vmatrix} a & 2b & 3c \\ 4a & 5b & 6c \\ 7a & 8b &9c \end{vmatrix}=a\begin{vmatrix} 1 & 2b & 3c \\ 4 & 5b & 6c \\ 7 & 8b &9c \end{vmatrix}=ab\begin{vmatrix} 1 & 2 & 3c \\ 4 & 5 & 6c \\ 7 & 8 &9c \end{vmatrix}=abc\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 &9 \end{vmatrix}$$