How to solve linear differential equations such as $y''-7y'+10y=0$? The question asks to solve the linear equation

$$y''-7y'+10y=0$$

I'm not sure where to even start.

I'm new to this so it would be nice if I got an in depth explanation ( with rules too )

Thank you!
I'm sure others will rush in and give you the solution to your particular equation.

Answering this question the way you want requires a whole lecture in ODEs. I recommend watching this good one from MIT on the exact topic, Solving second order linear ODEs with constant coefficients:

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-9-solving-second-order-linear-odes-with-constant-coefficients/

or equivalently

https://www.youtube.com/watch?v=vP-oRQqmeg4#t=30
Your differential equation has characteristic equation $t^2-7t+10=0$, which forces $t=-5,-2$. Therefore the solutions are of the form $f(x)=Ae^{5t}+Be^{2t}$.