The way I distinct them is by trying to get them into the two basic forms for separable and linear differential equations; that is

for linear and

for separable

For instance, in your examples;

The first,

[(1) say]

can be rewritten as

(assuming 1+x^2 can never be zero, in other words, no complex roots)

thus is of form

where

and

and so example (1) can be considered a linear differential equation.

The second,

[(2) say]

is much like (1) but, by writing y - 2xy as y(1-2x) you have a product of x terms and y terms (this is often - not always - a *hint* that the problem has a separable solution)

So, lets consider than the problem is separable, then the form should be like

that is

which then (by dividing by y and (1+x^2) - now watch since you're dividing by y there may be a singular solution at y=0 (or the general solution may include it) ) gives

then taking the integral of both sides with respect to x (you can think of this as multiplying each side by 'dx' and putting an integral sign on the front if its easier to remember - often it is) you get

thus (2) can be considered a separable differential equation.

Your third,

[3]

looks like it'd be a separable (mainly because of that x/2 factor) however quickly thinking over it would give a y/x term which could add more work than the alternative - which would be to rewrite this in linear form)

Now [3] can be written as

(taking y to the other side) and then,

(dividing by x/2, in other words multiplying by 2/x and not letting x be 0)

Now the above is in the general form for a linear differential equation where our f(x) and g(x) are

and

thus [3] can be solved as a linear differential equation

Finally [4],

well, right away (if you can notice that the entire right hand side is a product of x and y terms and the left hand side has only y' ) theres a great this differential equation is separable. So, take sqrt(y) over to the left to get (meaning there could (and likely is) a singular solution at y = 0 not in the general solution)

then integrating with respect to x on both sides gives

and so [4] can be solved by separating out the x-terms and y-terms.

Generally, in practice, with a lot of math, the best method tends to come from thinking about the problem because more often than not you'll get a differential equation that isn't in standard form and you'll need to look for the best method to use. Sometimes - though unwise - it's still a good idea to plough through the equation assuming it can be linear (for example) and you hit a dead end or a really hard equation in which case, turn back and assume it is separable - and hopefully you'll find a nicer looking expression to solve.

I hope some ideas of how make quicker deductions about an equation being separable or linear came from the above.

Edit:

Also, I though I might mention that there are occasions that even if you do get a nice looking, say separable, equation at the end - it may be impossible to solve in which case, the solution might lay with the linear equation - even if it looks incredibly 'horrible' and difficult to solve compared to the separable one at first.