What do 'general solution' and ODE mean? In an answer to this question, For what values of $r$ does $y=e^{rx}$ satisfy $y'' + 5y' - 6y = 0$?, URL (version: 2013-11-13): https://math.stackexchange.com/q/566059
$y' = [e^{rx}] (r)$
$y''= r^2e^{rx}$
It says:
If you plug them in, you obtain : $$r^2+5r-6=0$$
Solving this equation you get $r=1$ or $r=-6$.
That means that the general solution of the suggested ODE is : $$y(x)=ae^t + be^{-6t}, (a,b) \in \Bbb R^2$$
What do 'general solution' and ODE mean? Where does the last equation come from and what is its significance?
 A: A differential equation is an equation involving derivatives. An ODE or ordinary differential equation is a differential equation that contains one independent variable and its derivatives. $y'$ refers to the derivative of $y=f(x)$, $y''$ refers to the second derivative  of $y=f(x)$, etc. 
The general solution is the family of functions that satisfy the differential equation.
One thing we can do to find a solution is guess that a solution to the differential equation is of the form $y=e^{rx}$ and check to make sure that there does exist some $r$ such that the ODE will be satisfied:
$$y''+5y'-6y=0$$
Substituting in $y''$, $y'$, and $y$ gives:
$$r^2e^{rx}+5re^{rx}-6e^{rx}=0$$
Dividing by $e^{rx}$ on both sides gives an equation which is often called the characteristic equation:
$$r^2+5r-6=0$$
So $r=-6$ or $r=1$. We may check to make sure that $y=e^{-6x}$ and $y=e^{1x}$ are both solutions if we like. 
An equation of the form:
$$c_1y''+c_2y'+c_3y=0$$
Is called a $2nd$ order homogenous differential equation. It is simple to check that if $g(x)$ and $h(x)$ are both solutions to this differential equation then so is:
$$c_1g(x)+c_2h(x)$$
This accounts for a more general solution to the differential equation of interest. However proving this is the unique general solution if we want our solution to be continuous takes a bit more work. However, let me note that for a differential equation like a 2nd order home genius differential  equation there is a catch when the roots to the  characteristic equation are repeated. See this Repeated Roots
Or this video Khan Academy
By the way his previous differential equation videos I find pretty intuitive, it might be worth it to check them out.
