Confusion regarding the general solution theorem

Question

Theorem 1.13 (General Solutions)

If $y_1$ and $y_2$ are linearly independent solutions of the equation $L[y]=0$ on the interval $I\subset\mathbb R$ where $L[y]=y''+p(t)y'+q(t)y$ and $p(t),\ q(t)$ are continuous functions on $I$, then there exists unique constants $c_1,\ c_2$ such that every solution $y$ of the differential equation $L[y]=0$ can be written as a linear combination, $$ y(t) = c_1y_1(t) + c_2y_2(t).$$

I'm confused on the part which says that there exists unique constants $c1,c2$ such that every solution $y$ of the DE can be written as a linear combination. Shouldn't it be that for every solution $y$ there exists unique constants?

I've only recall DE's that have a unique solution called the particular solution for unique $c1,c2$. I feel like I'm missing something.

Answer 1

You are not missing anything. The statement should have been

" Every solution is a linear combination of $y_1$ and $y_2$ which means for each solution y(t),there are constants $c_1$ and $c_2$ such that $y(t)= c_1 y_1(t) + c_2 y_2(t)$"