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:(adsbygoogle = window.adsbygoogle || []).push({});

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 e^{rx}, 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 e^{rx}which does that?

Then I found a book where there was this theorem:

" If y_{1}and y_{2}are linearly independent solutions of equation, and P(x) is never 0, then the general solution is given by

y = c_{1}y_{1}+ c_{2}y_{2}

where c_{1}and c_{2}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 b^{2}= 4ac, you may assume that y = f(x)e^{rx}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)e^{rx}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.

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# Problem in understanding some theorems about second order linear differential equations

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