# How are these two theorems compatible?

I have the following two theorems:

Theorem 1: Consider the initial value problem $$y''+p(t)y' + q(t)y = g(t), y(t_0) = y_0, y'(t_0) = y'_0$$ where $p, q,$ and $g$ are continuous on an open interval $I$. Then there is exactly one solution $y = \phi(t)$ of this problem, and the solution exists throughout the interval $I$.

Theorem 2: Suppose that $y_1$ and $y_2$ are solutions of $$L[y] = y'' + p(t)y' + q(t)y = 0,$$ and that the Wronskian $$W = y_1y_2' - y_1'y_2$$ is not zero at the point $t_0$ where the initial conditions, $$y(t_0) = y_0, y'(t_0) = y_0',$$ are assigned. Then there is a choice of the constants $c_1,c_2$ for which $y = c_1y_1(t) + c_2y_2(t)$ satisfies the differential equation and the initial conditions.

Question: How can these theorems exist at the same time? Theorem 1 says that there is only one solution of the given initial value problem, while theorem 2 says that for the same (I'm obviously mistaken) initial value problem you can choose the constants $c_1$ and $c_2$. The only difference I see here is that $g(t) = 0$ in the second theorem. But if $g(t) = 0$ then $g(t)$ is still continuous so this shouldn't imply that in that case theorem 1 doesn't hold right?

Thanks!

• The choice of constants is unique, of course. :) – Pedro Tamaroff Feb 27 '18 at 12:01