Where can I find proofs of these theorems?

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In summary, there are three theorems in the conversation: one regarding uniqueness of solutions to a linear homogenous ODE, one known as Abel's Theorem, and one regarding the existence of a fundamental set for a linear homogenous ODE. The person asking for a proof of these theorems is recommended to refer to the text "Differential Equations" by George F. Simmons, which contains proofs for these theorems and many other topics.
  • #1
Bipolarity
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There are a few theorems in my DE book whose proofs I've been trying to find, without much luck:

1) Given an nth order linear homogenous ODE with continuous coefficients and nonzero leading coefficient, any IVP of this ODE has a unique solution over some interval I centered about the IVP.

2) Abel's Theorem for nth order linear homogenous ODE: Given an nth order linear homogenous ODE with continuous coefficients and nonzero leading coefficient, if [itex] \{y_{1},y_{2}...y_{n} \}[/itex] are all solutions to the ODE on some interval I, and they have derivatives up to order (n-1) on I, then their Wronskian is either 0 everywhere on I or nonzero everywhere on I.

3) Every nth order linear homogenous ODE with continuous coefficients and nonzero leading coefficient has a fundamental set.

In what textbook might I find a proof of these theorems? Thanks!

BiP
 
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  • #2
For the first one, I'd say suppose that y1(x) and y2(x) are both solutions to the IVP. Then consider y(x) = y1(x) - y2(x) and show that y=0 in which case you have uniqueness.
 
  • #3
Every DE book (that I've ever seen at least) has a proof of the fundamental theorem has the theorem: if f(x, y) is continuous in some neighborhood of [itex](x_0, y_0)[/itex] and f(x, y) is "Lipschitz" in y in some neighborhood of [itex](x_0, y_0)[/itex] ("differentiable" is stronger than "Lipschitz" and is used instead in many introductory texts) then there exist a unique y(x) satisfying [itex]dy/dx= f(x,y)[/itex] with condition to [itex]y(x_0)= y_0[/itex].

You can extend that to nth order equations, [itex]d^ny/dx^n= f(x, y, y', ..., y^(n-1)[/itex] by writing it as n first order equations:[itex]y_0= y[/itex], [itex]y_1= y'[/itex], [itex]y_2= y''[/itex], etc. and then writing that as the single vector equation [tex]dV/dx= F(x, V)[/tex] where [itex]V= <y_0, y_1, y_2, ..., y_{n-1}>[/itex].

2) I would be surprised if your text did not have that proof, at least in the second order case. Essentially, it involves showing that the Wronskian, W(x), itself satisfies a first order equation, which, if W(x) is 0 at any point, has only the identically 0 solution.

3) Again, I would expect to see this proof in any text. The point is that the derivatives themselves are "linear": d(Af+ Bg)/dx= A(df/dx)+ B(dg/dx). Give that, If y1 and y2 both satisfy the linear differential equation [itex]a_n(x)d^ny/dx+ a_{n-1}(x)d^{n-1}y/dx^{n- 1}+ \cdot\cdot\cdot+ a_0y= 0[/itex], then so does Ay1+ By2 for any numbers A and B. That is, the set of all such solutions forms a vector space. (You should always take Linear Algebra as a prerequsite to Differential Equations for exactly this reason.)

Now show that y1, satisfying that equation with y1(x0)= 1, y1'(x_0)= 0, y1''(x_0)= 0, ..., [itex]y1(n-1)(x_0)= 0[/itex], y2, satisfying that equation with y2(x_0)= 1, y2'(x_0)= 1, ...[itex]y2^(n-1)(0)= 0[/itex], ..., [itex]y^{(n-1)}(0)= 0[/itex], y3(x_0)= 0, y'(x_0)= 0, y'''(x_0)= 1, ..., [itex]y^{n-1}(x_0)= 0[/itex], etc gives n independent functions. Finally, show that if y satisfies the equation as well as the intial condition y(x_0)= A, y'(x_0)= B, y''(x_0)= C, ..., then y(x)= Ay0+ By1+ Cy2+... showing that these n functions form a basis for that vectors space- what you are calling a "fundamental set".
 
  • #4
Thanks Ivy, but could you recommend me a text that proves these theorems? My textbook proves neither of the three.

BiP
 
  • #5
Bipolarity said:
Thanks Ivy, but could you recommend me a text that proves these theorems? My textbook proves neither of the three.

BiP

That doesn't sound too good. The book https://www.physicsforums.com/showthread.php?t=665418 proves almost every theorem they use, is pretty easy to pick up from any point inside, and contains so many topics that you will probably use it as a reference in the future many times. It is also a Dover book, so it is extremely inexpensive compared to most other texts.
 

1. Where can I find proofs of these theorems?

There are several places where you can find proofs of theorems. One option is to look in textbooks or academic journals relevant to the subject. You can also search online for articles, lecture notes, or videos that provide proofs. Another option is to consult with a mathematician or professor who specializes in the area of the theorem.

2. Are all proofs of theorems available to the public?

Yes, most proofs of theorems are available to the public. However, some may be behind paywalls in academic journals or require a subscription to access. In these cases, you may need to visit a university library or reach out to the author directly to obtain the proof.

3. Can I trust proofs of theorems found online?

It is always important to verify the credibility of any source, including proofs found online. Make sure the source is reputable and that the proof has been peer-reviewed by experts in the field. Additionally, it is a good idea to compare the proof to others found in textbooks or academic journals to ensure its accuracy.

4. How can I understand a proof if I am not an expert in the subject?

If you are not familiar with the subject of the theorem, it may be helpful to first gain a basic understanding of the concepts and terminology involved. You can do this by reading introductory textbooks or watching online lectures. It may also be beneficial to seek guidance from a mathematician or professor who can explain the proof in simpler terms.

5. What should I do if I cannot find a proof for a particular theorem?

If you are unable to find a proof for a specific theorem, you can try reaching out to mathematicians or professors who specialize in the area of the theorem. They may be able to provide you with a proof or direct you to relevant resources. Additionally, you can try to approach the theorem from a different angle or consult with colleagues or peers for insights and ideas.

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