Differential Equations: Wronskian question.

In summary, the conversation is about a student asking for help with solving a problem involving the Wronskian and showing linear independence of functions. They are looking at problem 34 and also mention problem 33. Another student responds with a hint and suggests looking at the Wronskian for the n=3 case and using a property for solving determinants. The original student expresses their understanding of where the matrix comes from but asks for clarification on the method for solving the determinant. The other student suggests using a specific property and provides an example.
  • #1
tarmon.gaidon
31
0

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.

000004.jpg


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
 
Last edited:
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  • #3
Mark44 said:
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.
 
  • #4
Let's start by looking at 33. Do you know how to use the Wronskian to show that functions are linearly independent in general?
 
  • #5
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:

8d606b824d0946483b111ce8935ba568.png


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.
 
  • #6
tarmon.gaidon said:
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:

8d606b824d0946483b111ce8935ba568.png


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 [itex]f_i(x)=\text{exp}(r_i x)[/itex], [itex]1\leq i \leq n[/itex], what is [itex]f'_i(x)[/itex]? How about [itex]f''_i(x)[/itex]? What does that make the Wronskian for the [itex]n=3[/itex] case? Is there a rule for taking determinants where a column is multiplied by some factor that might help you here?
 
  • #7
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?
 
  • #8
Use property 3 here. For example,

[tex]\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}[/tex]
 

Related to Differential Equations: Wronskian question.

1. What is a Wronskian in differential equations?

A Wronskian is a mathematical tool used to determine the linear independence of a set of functions. In the context of differential equations, it is used to check if a set of solutions to a linear homogeneous equation are linearly independent.

2. How is a Wronskian calculated?

The Wronskian of a set of functions is calculated by arranging the functions in a matrix and taking the determinant of that matrix. For example, if the functions are y1(x) and y2(x), the Wronskian would be calculated as:
W = |y1(x) y2(x)|
|y1'(x) y2'(x)|

3. What is the significance of the Wronskian in differential equations?

The Wronskian is an important tool in determining the general solution to a non-homogeneous linear differential equation. It can also be used to determine the existence of a fundamental set of solutions for a homogeneous equation.

4. Can the Wronskian be used for non-linear equations?

No, the Wronskian is only applicable for linear differential equations. It cannot be used for non-linear equations because the linearity property is necessary for the Wronskian to be a useful tool.

5. How does the Wronskian relate to the concept of linear independence?

The Wronskian is used to test for linear independence by determining if the determinant of the matrix formed by the functions is equal to zero. If the determinant is non-zero, then the functions are linearly independent. However, if the determinant is equal to zero, then the functions are linearly dependent.

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