Htykuut

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?

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".

To find a solution ''by inspection'' usually means that the solution can be found (almost) immediately, using some result that is expected to be well known.

In your case, as an example, from the fact that the derivative of $y=x^2$ is $y'=2x$ we can say ''by inspection'' that the solution of $xy'=2y$ can be a function of the form $y=x^2$, that satisfies the initial condition.

Analogously, using the fact that $y=x^3 rightarrow y'=3x^2$ you can solve ''by inspection'' the other IVP.

I already see that the trivial solution $y=0$ satisfies both systems. We can slso try $y=x^3$ for the first and $y=x^2$ for the second.