Rudin Ch 5, #29: Uniqueness Theorem for Systems of ODEs

[Quantumpencil]

Homework Statement

Specialize exercise 28 by considering the system

[tex]\y'= y_{j+1} j=(1,...,k-1)
y'_{k}= f(x)-\sum g_{j}(x)y_{j} where the summation runs from j=1 to j=k, and g_{j} and f are continuous real functions on [a,b], and derive a uniqueness theorem for solutions of the equation

y^{k}+g_{k}(x)y^{k-1}+...+g_{2}y'+g_{1}(x)y = f(x)

subject to initial conditions

y(a)=c_{1}, y'(a)= c_{2}, y^{k-1}(a) = c_{k}.[\tex]

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi Quantumpencil!

(you use the wrong slash: )

Specialize exercise 28 by considering the system

$$y'= y_{j+1}, j=(1,...,k-1),\ y'_{k}= f(x)-\sum g_{j}(x)y_{j}$$ where the summation runs from j=1 to j=k, and g_{j} and f are continuous real functions on [a,b], and derive a uniqueness theorem for solutions of the equation

$$y^{k}+g_{k}(x)y^{k-1}+...+g_{2}y'+g_{1}(x)y = f(x)$$

subject to initial conditions

$$y(a)=c_{1}, y'(a)= c_{2}, y^{k-1}(a) = c_{k}$$

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[Quantumpencil]

yeah, so I didn't realize I made this thread; it's incomplete. Could I get it locked?

The actual thread I need help on is further-down and contains the good tech + information about how I think the solution will work out.

https://www.physicsforums.com/showthread.php?t=297047

This is the link to the actual topic.