I don't understand 'variation of parameters' I am currently reading my notes on the 'variation of parameters' method to solve differential equations, and I admit that it has left me nothing but discombobulated. 
So this is what my note say:
$$\dfrac{dy}{dt} - cy = -ck$$
So apparently this is the DEQ we want to solve. The next step is $$\dfrac{dy}{dt} - cy = 0$$
So I'm guessing this is a standard step in the procedure? $$y = be^{ct}$$
This I understand, but the next steps puzzle me, here are they $$y=u(t)e^{ct}$$ $$\dfrac{dy}{dt} = \dfrac{du}{dt} e^{ct} + u(t) ce^{ct}$$ 
And then some more steps, but at this juncture I am already too confused to continue. What is actually done here? How does this method actually work? I've tried looking it up but most explanations are just too formal and I need a more 'applied' explanation, less rigour (I'm sorry mathematicians).
 A: Most of these 'puzzling' steps are just clever guesses that wind up working out. Remember that one of the main methods for solving differential equations is making a good guess for a particular type of solution and then refining that guess to a specific answer.
In particular, when you want to solve a non-homogeneous equation
$$
\frac{dy}{dt} - cy = -k \quad \quad (*)
$$
(non-homogeneous b/c right-hand-side is non-zero), we take inspiration from the corresponding homogeneous equation
$$
\frac{dy}{dt} - cy = 0.
$$
We know the solution to the homogeneous equation is $be^{ct}$, and so we guess that a solution to the non-homogeneous equation might be $u(t)e^{ct}$ (the $b$ in the homogeneous solution is constant, we are sort of guessing that the additional complication of a non-homogeneous term might lead to the additional complication of the term in front of $e^{ct}$ no longer being constant).
Now, we see what the implications of this guess are, in the hopes that we can determine $u(t)$. In particular
$$
\frac{d}{dt}(u(t)e^{ct}) = u'(t)e^{ct} + cu(t)e^{ct}
$$
so if this is going to be a solution of $(*)$ (i.e. if we plug it into the original diff. eqn.), then we need
$$
u'(t)e^{ct} + cu(t)e^{ct} - cu(t)e^{ct}= -k \\
u'(t)e^{ct} = -k
$$
$$
u'(t) = -ke^{-ct}, \quad \quad (**)
$$
and now we have reduced the problem to solving a differential equation for $u$.
That is, if we can can solve the equation $(**)$, then we will be sure that $u(t)e^{ct}$ is a solution of $(*)$.
