Question: Given two matrices $A\_{m\times n}$ and $B\_{n\times p}$, what is the sufficient and necessary condition for $AB$ to have full rank?

I know $r(AB)=r(B)-\dim N(A) \cap R(B)$, so is it true the above iff $\dim N(A)\cap R(B)=0$? But seems incorrect if $m\lt p$. Please help.

Answer: 
The indefinite integral $\int f(x) dx$ denotes, in this particular context, the *set* of all primitives of $f(x)$. If you find one primitive, say $F(x)$, then you know all other primitives have the form $F(x) + C$, where $C$ is any constant. This is what we mean when we write $\int f(x) dx = F(x) + C$: the set of all primitives of $f(x)$ is the set of all the functions of the form $F(x) + C$.
The indefinite integral $\int f(x)\,dx$ is defined to be the general class of *functions* whose derivatives are $f(x)$. 

Now, given any function $F$ with $F'=f$, it follows that $F+C$ is also an antiderivative of $f$:
$$
(F+C)'=F'+C'=F'+0=F'=f.
$$
Conversely if $G$ is an antiderivative of $f$, then $G$ has the form $F+C$ for some constant $C$.

That's why we write, for example:
$$
\tag{1}
\int x^2\,dx={x^3\over 3}+C.
$$
The derivative of ${x^3\over 3}+C$ is $x^2$. Moreover if $F'(x)=x^2$, then $F$ must have the form $F(x)={x^3\over3}+C$.

So, the general antiderivative of $f(x)=x^2$ has the form $F(x)={x^3\over3}+C$, and equation (1) is just stating this fact.

The differential equation you are considering has more than one solution. The *general solution* is as you have, with the arbitrary parameter $C$. It does belong there. 
Any function of that form will be a solution to the equation.

If you were given an initial value problem where, say, you needed $y(1)=2$, then that constant $C$ could be determined.