How to prove this theorem about linear ODE's?

In summary: If it's not linear, then I'm not sure what to do.In summary, the author is proposing a method for finding linearly independent solutions to an n-1th order linear d.e. However, the method has many problems.
  • #1
AdrianZ
319
0

Homework Statement


Suppose that we have the following linear ODE:
[tex]y^{(n)} + a_{n-1}y^{(n-1)} + a_{n+2}y^{(n-2)} + ... + a_2y'' + a_1y' + a_0y = 0[/tex]
Prove that if we have (n-1) linearly independent {y1,...,yn-1} solutions then we can find yn.

The Attempt at a Solution


well, this is my idea:
Imagine that C(a,b) is the linear space of all the functions that are continuous in the interval [a,b]. Let's define an inner product on this space as the following:
<f,g> = ∫abf(x)g(x)dx
Now, Imagine I have n-1 linearly independent functions y1,...,yn-1 then the problem is transformed into finding a function yn such that yn is perpendicular to all the other yi's (i=1,...,n-1).

well, it means I should show that there exists a function that for any yi I have:
abynyidx

Here is where I'm stuck. What should I do?
 
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  • #2
I'm not sure what you mean by "find [itex]y_n[/itex]". Certainly it is true that if you know n-1 (independent) solutions to an nth order linear d.e., you can reduce to a first order equation for the nth independent solution. But, of course, you may not be able to solve that first order equation!

I see many problems with your proposed method. For example, you talk about "functions continuous on [a, b]". Where did "a" and "b" come from? There is no such interval given in the problem nor is there any reason to think a solution would be defined only on such an interval.
 
  • #3
HallsofIvy said:
I'm not sure what you mean by "find [itex]y_n[/itex]". Certainly it is true that if you know n-1 (independent) solutions to an nth order linear d.e., you can reduce to a first order equation for the nth independent solution. But, of course, you may not be able to solve that first order equation!
well, would you tell me how I can reduce the equation to a first order equation for the nth independent solution? Would that new ODE be linear? If yes, then why I may not able to solve that first order ODE?
Anyways, your idea seems nice. Please tell me how I can reduce my equation to a first ODE if I have (n-1) independent solutions.

I see many problems with your proposed method. For example, you talk about "functions continuous on [a, b]". Where did "a" and "b" come from? There is no such interval given in the problem nor is there any reason to think a solution would be defined only on such an interval.
well, you're right but a and b are not my problem right now to be honest because the inner product I've defined stays well-defined even If I define <f,g> = ∫f(x)g(x)dx. Am I right?
My idea was naive. I thought we have a nth order linear ODE and we know we have (n-1) vectors (linearly independent solutions), so If I find a vector (in this case a continuous function) that is perpendicular to all of them, then it surely is linearly independent and I'll have a basis for the linear space of its solutions. That was my idea.
 

1. How do I prove a linear ODE theorem?

To prove a linear ODE theorem, you will need to use mathematical techniques such as substitution, integration, and differentiation. It is also important to fully understand the properties and rules of linear ODEs before attempting to prove a theorem.

2. What are the key steps in proving a linear ODE theorem?

The key steps in proving a linear ODE theorem include setting up the ODE, applying mathematical techniques to simplify the equation, and using logical reasoning to show that the theorem holds true for all possible solutions.

3. Can I use examples to prove a linear ODE theorem?

Yes, using examples can be a helpful tool in proving a linear ODE theorem. However, it is important to also provide a logical and mathematical explanation for why the theorem holds true, rather than just relying on examples.

4. What are some common mistakes to avoid when proving a linear ODE theorem?

Some common mistakes to avoid when proving a linear ODE theorem include incorrect application of mathematical techniques, not considering all possible solutions, and making assumptions without proper justification.

5. How do I know if my proof of a linear ODE theorem is correct?

A correct proof of a linear ODE theorem should be logically sound, use proper mathematical techniques, and provide a clear explanation for why the theorem holds true. It is also helpful to have someone else review your proof for any errors or inconsistencies.

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