Show that $y_1$ and $y_2$ are not Linearly Independent

Suppose $y_1$ and $y_2$ vanish at $t_0$. Then both functions satisfy the initial value problem $y''+py'+qy=0$ with $y(t_0)=0$.

If we complement the initial-value problem with an initial value of the type $y'(t_0)=c$, $c\in\mathbb R$, then the equation is uniquely solvable.

Moreover, we observe that the mapping $S:c\mapsto y$ is linear, where $S$ is the solution mapping of the differential equation, mapping the initial value $y'(t_0)=c$ to the whole trajectory.

Now, $y_1'(t_0)$ and $y_2'(t_0)$ are values in the one-dimensional vector space $\mathbb R$, thus they are linearly dependent. Hence their images under $S$ - the trajectories $y_1$ and $y_2$ - are linearly dependent.