I'm currently viewing a textbook on differential equations. They were demonstrating a introductory example on reduction of order where they derived a general solution
The question was:
Given that $y_1 = e^x$ is a solution of $y''-y=0$ on the interval $(-\infty, \infty)$, use reduction of order to find a second solution $y_2$.
Eventually, they got to
$$y=u(x)e^x=-\frac{c_1}{2}e^{-x}+c_2 e^x.$$
But then they said "by picking $c_2=0$ and $c_1=-2$, we obtain the desired second solution $y_2=e^{-x}$."
Question. How come in this case, we can choose particular values for the integration constants? Sometimes, when we are given initial conditions for different problems the integration constants are not free parameters, but are fixed constants that need to be solved for. Does that mean this method (reduction of order) is not valid for an IVP?
Link for screenshot from textbook:
Your answer is here
hence the general solution is $$y=c_1 e^x+c_2 e^{-x}$$
**************
In addition you can also follow the comment of Inn.