Vector space for solutions of differential equations

[Trying2Learn]

TL;DR Summary

Why is the solution of a diff.eq. a vector space

Good Morning

Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you.

In it, someone wrote:

" It is a theorem in mathematics that the set of all functions that are solutions of a linear differential equation is a vector space , sub space of the vector space of all functions (of a real variable). "

Is there a chance someone could provide the name of this theorem and provide link (preferably on-line) to a simple, introductory discussion about it?

 

[wrobel]

I believe that this fact is noticed in all textbooks. But you hardly find a proof because it is too trivial.

 

[Trying2Learn]

I believe that this fact is noticed in all textbooks. But you hardly find a proof because it is trivial.

Could you provide a reference... I just need to see it stated and the context.

I am a mech.eng. with flawed appreciation for math. I use it, machine like, and would like to explore this.

It is NOT trivial for me.

 

[wrobel]

http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf

page 81

 

[martinbn]

It is NOT trivial for me.

To be a vector space means that if you have two solutions, say $y_1(x)$ and $y_2(x)$, then any linear combination of them is also a solution. This means that any functions of the form $h(x)=ay_1(x)+by_2(x)$, where $a$ and $b$ can be any real numbers is also a solution to your equation.