Here's the problem I have been given:
Use the method of variation of parameters to find a general solution of the following differential equation:
$$ a_1y''+a_2y'+a_3y=f(x) $$
And linearly independent solutions $y_1$ and $y_2$ are known.
Note: $a_1, a_2,$ and $a_3$ are constants.
Assume that $y_p$ is of the form $u_1y_1+u_2y_2$ where $u_1, u_2$ are functions.
Now to solve the system of equations obtained by differentiating $y_p$: $$ \begin{cases} u_1'y_1+u_2'y_2=0 \\ u_1'y_1'+u_2'y_2'=f(x) \end{cases} $$
to solve the first equation for $u_1'$, we get $$ u_1'=-\frac{u_2'y_2}{y_1} $$
My question is how we know that $y_1$ cannot be zero depending on the value of the given independent variable $t$, and if it doesn't matter, why?