I have two questions regarding solving differential equations given initial conditions:
1) When do you substitute the initial conditions into the equation to calculate the value of the constant "$c$". Do you substitute it once you integrate both sides of the differential equations and you get a constant "$c$"? Or do you substitute the initial conditions after integrating both sides AS WELL AS rearranging the equations to get $y$ in terms of $x$ and $c$. Using the second method, sometimes you get two values for "$c$" with only one value being correct.
2) When you solve certain differential equations, you get one side written with "$\pm$" in the front. However, only one equation fits the initial conditions even after you solve for the constant "$c$". The one that fits is either the one with the "$+$" or the one with the "$-$" in the front. How do you justify which one is correct without giving geometric representations of both and then saying "according to graph, this one insert equation is correct". Can you somehow solve without getting the "$\pm$" in the front?