So, I had studied oscillatory motion for a while and I found it unpleasant to have to remember the various different solutions for the equations of motion. I began to learn about second order linear differential equations and now I know how to solve this kind of stuff. But there is a problem: For the next considerations, this is the general form of the equation: P(x)y'' + Q(x)y'+ R(x)y = 0 , where P,Q,R are functions of x Firstly, I watched some videos and that guy said there is theorem that says the solution it's always formed by erx, where r is a constant (but he gave no proof). And by replacing this function in the equation you can find r and then form the general solution. My first problem is: how do you show that there is no other function except the erx which does that? Then I found a book where there was this theorem: " If y1 and y2 are linearly independent solutions of equation, and P(x) is never 0, then the general solution is given by y = c1y1 + c2y2 where c1 and c2 are arbitrary constants." But the authors give no proof. So my second problem is how do you actually prove this theorem. The next questions are related to the case P(x)=a, Q(x)=b, R(x)=c, where a,b,c are real constants. If you have b2 = 4ac, you may assume that y = f(x)erx is a solution and you will get that f''(x)=0. Now it is obvious that f(x) = cx +c' satisfies f''(x)=0, where c and c' are some constants. So here are two questions: 1. Why you take y = f(x)erx and not other function? 2. Is f(x) = cx +c' the only solution for f''(x)=0? I'd also be very grateful if someone would suggest a book on differential equations that really studies them deeply.