Discussion problem, wronskian matrix, linear independence of solution

In summary, the conversation discusses a problem involving linearly independent solutions to a homogeneous differential equation. The question is whether the Wronskian matrix for the solutions is equal to zero for every real number x. It is argued that this violates a theorem which states that the set of solutions is linearly independent if the Wronskian is not equal to zero for every x in the interval. However, it is pointed out that the theorem also states that the leading coefficient of the differential equation must be nonzero, which is not the case in this situation. This leads to a discussion about the importance of explicit statements and assumptions in theorems.
  • #1
AdkinsJr
150
0
Edit: I think I may have posted this in the wrong section, sorry about that. Note that this isn't a homework problem though, I"m not enrolled in this class, I was just reading over some of this stuff and trying some problems since I"m majoring in physics.

I have a textbook "discussion" problem that's stumping me, I'm given that these are two linearly independent solutions:

[tex]y_1=x^3[/tex] and [tex]y_2=\mid x\mid ^3[/tex]

to the homogeneous differential equation [tex]x^2y''-4xy'+6y=0[/tex]

on [tex](-\infty,\infty)[/tex]

I'm asked to show that the wronskian matrix for the solutions is equal to zero for every real number x. This is easy enough to do, I won't show that work here... but then I"m asked whether this violates the theorem for the wronskian matrix test, which is stated in my text:

If [tex]y_1,y_2...y_n[/tex] are n solutions of a homogenous linear nth-order differential equation on an interval I. Then the set of solutions is linearly independent on I if and if [tex]w(y_1,y_2...y_n)≠0[/tex] for every x in the interval. Naturally I would say yes, it is a violation, but they seem to be implying that it isn't a violation (i can just tell by the way they're asking...). They're messing with me. Why is it not a violation?
 
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  • #2
If you look at the theorem where it is proven that the Wronskian is either never zero or else identically zero, you will find that one of the hypotheses is that the leading coefficient of the DE is nonzero. If you look at the proof, you will see why. This equation fails at ##x=0## and the result may fail on any interval containing zero. Notice, however, on any interval not containing zero it all works.
 
  • #3
LCKurtz said:
If you look at the theorem where it is proven that the Wronskian is either never zero or else identically zero, you will find that one of the hypotheses is that the leading coefficient of the DE is nonzero. If you look at the proof, you will see why. This equation fails at ##x=0## and the result may fail on any interval containing zero. Notice, however, on any interval not containing zero it all works.

thanks, wouldn't that usually be have stated in the theorem though? usually with theorems in textbooks I don't find there is anything crucial missing. You think the theorem would state that it should be a linear homogeneous equation with [tex]a_n(x)≠0[/tex] for x in I...or something like that, if I'm understanding correctly. My text doesn't provide a proof so I'll check that out online somewhere.
 
  • #4
I think you will find that ##a_n(x)\ne 0## is explicitly stated or the leading coefficient is ##1##. In that case the hypothesis is inherent. You can always divide both sides of the DE by ##a_n(x)## if it is nonzero to get a leading coefficient of ##1##.
 

1. What is a Discussion Problem in the context of linear algebra?

A Discussion Problem in linear algebra refers to a question or topic that requires critical thinking and analysis to solve. These problems often involve concepts such as matrices, systems of equations, and linear transformations.

2. What is a Wronskian Matrix and how is it used?

A Wronskian Matrix is a square matrix that is formed by taking the derivatives of a set of functions. It is used to determine the linear independence of a set of solutions to a differential equation. If the determinant of the Wronskian Matrix is non-zero, then the solutions are linearly independent.

3. How is the linear independence of a solution determined using the Wronskian Matrix?

To determine the linear independence of a set of solutions using the Wronskian Matrix, calculate the determinant of the matrix. If the determinant is non-zero, then the solutions are linearly independent. If the determinant is zero, then the solutions are linearly dependent.

4. Can the Wronskian Matrix be used to determine the general solution of a differential equation?

No, the Wronskian Matrix can only be used to determine the linear independence of a set of solutions to a differential equation. To find the general solution, the Wronskian Matrix can be used in combination with other methods such as variation of parameters or the method of undetermined coefficients.

5. How does the linear independence of solutions relate to the fundamental set of solutions in a differential equation?

The fundamental set of solutions in a differential equation is a set of linearly independent solutions that form the basis for all other solutions. Therefore, the linear independence of solutions is crucial in determining the fundamental set of solutions and ultimately, the general solution to the differential equation.

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