Prove that the dimension of the solution space of a second order homogenous ODE is 2

Question

I know that questions like this have been asked here before, but they rarely answer my question. Here, we consider \begin{align}y''+by'+cy=0\end{align} I want to prove that the dimension of the solution space of a second order homogenous ODE is 2.

MY TRIAL:

My instructor told us to reduce the above equation to a system of first-order odes. So. if we let $z=y'$, then \begin{align}\begin{cases}y'=z, &\\z'=bz+cy&\end{cases}\end{align} Question:

How do we have the dimension of the solution space is 2 from here ?

Answer 1

Using the Existence and Uniqueness theorem, the solutions $(y,z)$ are uniquely specified by their initial values $(y(0), z(0))$. The solution with initial values $y(0)=1, z(0)=0$ and the solution with initial values $y(0)=0,z(0)=1$ form a basis of the solution space.