On the idea of a general solution to an ODE I am using the book A Textbook on Ordinary Differential Equations
 by Shair Ahmad and Antonio Ambrosetti, on ODEs. After reading the first two chapters I got stuck on the idea of a general solution of an ODE. 
In the first chapter the authors begin by considering Malthus equation and say that
$x(t)$ satisfies the equation if and only if $x(t)=ce^{kt}$ for some $c$ and is thus a general solution.
On the other hand when looking at the general case of integrating factors they say the following

"We see that if $x(t)$ satisfies the equation then there is a $c$ such that $x$ has the form ...  Moreover it is easy to see that for all $c$, $x$ of the form ... satisfy the equation. This is why $x$ is called a general solution.

Is there a "general" idea of a general solution or do they differ depending on the equation?
 A: Usually, and ODE comes together with some boundary conditions and/or initial conditions. Under the right assumptions, the ODE then has a unique solution satisfying the conditions.
As long as you don't specify additional conditions, an ODE will usually have a whole family of solutions, which can be parametrized by one (or more) parameter $c$. Only after imposing the boundary/initial conditions you can determine the parameter $c$, such that your function solves the ODE and the imposed conditions.
Essentially, a solution still containing the unspecified constants due to integration of the ODE is a general solution. But strictly speaking, it is not one, single solution, but a family of solutions, parametrized by the constants. As soon as you fix $c$, you have a solution of the ODE, and no longer a general solution.
A crucial point is that every solution of the ODE can be obtained from the general solution, by fixing the constant accordingly.
A: I think it is only the classical terminology of the theory of differential equations. Instead of speaking of the set of all the solutions of the equation, one speaks of the general solution because for simple equations, the general solution can be described by a formula containing a few arbitrary constants.
In the same way, you could say that a general plane in 3 dimensional space has the equation $a x + b y + c z = d$ where $a, b, c, d$ are real constants. It's easier than speaking about the set of all the planes. 
