Properties of Ordinary Differential Equations solutions

[Like Tony Stark]

I want to see if I've got clear some concepts regarding differential equations.
Let's suppose we have two solutions of a linear homogenous differential equation. Are these solutions always linearly independent, orthonormal and analytical?
Well, correct me if I'm wrong:
I think that they may not be linearly independent: we can consider a solution Y1 multiplied by some constant C1, and then the same solution multiplied by a constant C2. These are not L.I since they just differ by a constant. This also means that these solutions don't form a vector space.
In the case of a linear homogenous ODE, the solutions are always analytical since they are given by exponential or trigonometric functions, which don't have singular points.
As for the orthonormality I don't know what to say

 

[LCKurtz]

Sort of a confusing post, but...

I want to see if I've got clear some concepts regarding differential equations.
Let's suppose we have two solutions of a linear homogenous differential equation. Are these solutions always linearly independent, orthonormal and analytical?

I think you mean "analytic".

Well, correct me if I'm wrong:
I think that they may not be linearly independent: we can consider a solution Y1 multiplied by some constant C1, and then the same solution multiplied by a constant C2. These are not L.I since they just differ by a constant.

True.

This also means that these solutions don't form a vector space.

Not clear exactly what you are claiming, but probably false. The set of all constants times a single function forms a 1 dimensional vector space. That has nothing to do with whether they are solutions of any DE.

In the case of a linear homogenous ODE, the solutions are always analytical since they are given by exponential or trigonometric functions, which don't have singular points.

True for constant coefficient linear homogeneous DE's. Not in general.