I don't know what I am doing when I solve differential equations using the integrating factor and why does this give us the solutions it does?

Start by looking at the form of equation where we can use the integrating factor:y' + f'(x)y = g(x) The general principle of this method is that by multiplying our expression by some function of x it becomes far easier to determine the function y.Start from the position that in theory if something is just a function of x then it is possible to integrate it, this means that so long as both sides of our expression are only ever manipulated by functions of x then the right hand side ( originally g(x) ) will always be integratable. The overal aim of this method is to use this idea (the fact that we can change both sides of the equation in any way so long as we use functions of x) to get something of the form d/dx (yh(x)) = m(x) which may also look like y'h(x) + yh'(x) = m(x)Then this would be extremely useful, as simply integrating both sides of our expression would get us toy*h(x) = integral of m(x)This is precisely what is happening when we multiply by our integrating factor our initial expression from earliery' + f'(x)*y = g(x) ( integrating factor is e^{integral of f'(x)})becomes e^f(x)*y' + f'(x)e^f(x)*y = g(x) or even better d/dx ( e^f(x)*y) = g(x) now you can see why i chose to use f'(x) earlier, this is a completely fine thing to do despite it not being the form of the equation that is often shown.From this point it becomes very easy to find the function y.