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Where can I find proofs of these theorems?

  1. Mar 20, 2013 #1
    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
     
  2. jcsd
  3. Mar 20, 2013 #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.
     
  4. Mar 20, 2013 #3

    HallsofIvy

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    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".
     
  5. Mar 20, 2013 #4
    Thanks Ivy, but could you recommend me a text that proves these theorems? My textbook proves neither of the three.

    BiP
     
  6. Mar 21, 2013 #5
    That doesn't sound too good. The book "Ordinary Differential Equations" by Tenenbaum and Pollard 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.
     
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