difference between implicit and explicit solutions?

As requested:

Let's use the example initial-value problem

$$y^\prime y=-x,\qquad y(0)=r, \qquad r\text{ constant}$$

One can derive both an implicit and explicit solution for this DE. The implicit solution to this DE is

$$x^2+y(x)^2=r^2$$

This solution implicitly defines $y(x)$; all we have here is an equation involving $y(x)$. On the other hand, the explicit solution looks like

$$y(x)=\pm\sqrt{r^2-x^2}$$

and in this case, $y(x)$ is explicitly defined: $y(x)$ is expressed here as an explicit function with $x$ as the only independent variable.

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We aren't always this lucky when we solve differential equations that show up in practice. It often happens that we can only be content with an implicit solution (or a parametric solution, which is a somewhat better state of affairs than having just an implicit solution). One famous example is the differential equation that pops up in the brachistochrone problem:

$$(1+(y^\prime)^2)y=r^2$$

Explicit solution is a solution where the dependent variable can be separated. For example, $x+2y=0$ is explicit because if y is dependent, I can rewrite it as $y=-\frac{x}{2}$ and my y has been separated.

Implicit is when the dependent variable cannot be separated like $\sin(x+e^y)=3y$.