Where can I find proofs of these theorems?

[Bipolarity]

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 $\{y_{1},y_{2}...y_{n} \}$ 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

 

[Marioeden]

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.

 

[HallsofIvy]

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 $(x_0, y_0)$ and f(x, y) is "Lipschitz" in y in some neighborhood of $(x_0, y_0)$ ("differentiable" is stronger than "Lipschitz" and is used instead in many introductory texts) then there exist a unique y(x) satisfying $dy/dx= f(x,y)$ with condition to $y(x_0)= y_0$.

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

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 $a_n(x)d^ny/dx+ a_{n-1}(x)d^{n-1}y/dx^{n- 1}+ \cdot\cdot\cdot+ a_0y= 0$, 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, ..., $y1(n-1)(x_0)= 0$, y2, satisfying that equation with y2(x_0)= 1, y2'(x_0)= 1, ...$y2^(n-1)(0)= 0$, ..., $y^{(n-1)}(0)= 0$, y3(x_0)= 0, y'(x_0)= 0, y'''(x_0)= 1, ..., $y^{n-1}(x_0)= 0$, 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".

 

[Bipolarity]

Thanks Ivy, but could you recommend me a text that proves these theorems? My textbook proves neither of the three.

BiP

 

[slider142]

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.