Q:

Determine a solution of an ODE by "inspection"

I'm checking Zill's A First Course in Differential Equations with Modeling Applications, and there's an exercise that says:

From the following problems determine by inspection at least two solutions of the given IVP.

$y'=3y^{2/3},\,y(0)=0$
$xy'=2y,\,y(0)=0$

I don't quite understand what in means by "by inspection", what's the difference between just finding the solutions and determine by inspection?

A:

In general, the phrase "by inspection" means "by reasonable guessing". For instance, suppose we are to prove that there are integers $x,y,z$ such that $x^{2}+y^{2}=z^{2}$. Without manipulating the given condition, we have a candidate solution to the equation in mind, i.e. the triple $(3,4,5)$ of integers, which happens to be a solution to the equation; hence the proposition is proved. Well, we just proved the proposition by inspection. 
Yes, it is just the same thing to say "just find some solutions" instead of "find some solutions by inspection". 

