# Confusion regarding the general solution theorem

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.

• Yes, for every solution, there exist unique $c_1$ and $c_2$. Not the other way round – Shubham Johri Dec 11 '18 at 20:26

" 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 $$y(t)= c_1 y_1(t) + c_2 y_2(t)$$"