What is the philosophy behind solving ODE's? (a question from someone who hasn't taken a formal ODE's class) I've started learning ODE's on my own and here is something that I don't understand. I've noticed that the book I am following (and all the other books that I have) is hand wavy when it comes to specifying the interval of the solution and doesn't realy worry too much about dividing by $0$. I will provide an example: let's solve the ODE $$t^2x'=x^2+tx+t^2,$$ where $x=x(t)$.
We divide by $t$ and the equation becomes $x'=\left(\frac{x}{t}\right)^2+\frac{x}{t}+1$. We make the variable change $y=\frac{x}{t}$ and after some computations we get that $\arctan y=\ln t+C$ for some constant $C\in \mathbb{R}$. Now, the book says that this implies that $y=\tan(\ln t+C)$, so $x=t\tan(\ln t+C), C\in \mathbb{R}$. I have two questions here:

*

*Why can we divide by $t$ at the beginning? I mean, yes, I agree that this solves our equation, but aren't we kind of missing some solutions? Here is the first philosophical problem that I have with ODE's: is the focus on somehow obtaining a solution, even though we make some assumptions along the way, that is defined on some interval $I\subset \mathbb{R}$ that we don't even care if it is really really small rather than on trying to find all the differentiable functions that satisfy our identity (as the focus was in, say, functional equations that appear at high school math contests)?

*Why after $\arctan y=\ln t+C$ we may write that $y=\tan(\ln t+C)$ for any real constant $C$? I mean, the $\tan$ function is not defined everywhere and we most certainly can choose some $C$ such that for some $t$ we have $\ln t+C=\frac{\pi}{2}$ for instance. Is the philosophy here the same that I presented in 1 i.e. assuming that the interval on which our solution is defined is chosen appropriately so that everything makes sense?

 A: Whenever you manipulate any equation of any kind, you ought
to say first, "Suppose there is a solution" because of course
some problems have no solution but you won't know until you
have researched. In your example, it could be
Suppose there is a differentiable function defined on some interval
not containing zero. This is often not stated in introductory texts.
After all, the real test of any proposed solution formula is
not what you might have written on scratch paper to obtain it,
but to plug into the ode and check that it really makes sense
and is a solution having whatever initial or boundary values
you were looking for.
Also, once you study the Fundamental Existence Theorem for ODEs
you find out that solutions of initial value problems are only
guaranteed to exist for possibly short intervals around the
initial condition. Your example illustrates this. In fact the
Fundamental Theorem is about equations $x' = f(x,t)$ which
only applies to your example after you exclude $t=0$ and divide.
I think this covers both your questions.
I'll add a further note about what you might call philosophy.
ODE is an old subject going back to Newton and others who solved
sometimes very difficult equations, most of which had serious
applications to science, and were done before "number" and "function"
were even defined in the modern sense. Some of the early methods
and terminology have been retained for 350 years in texts, so
they can sound pretty odd if you compare them to your abstract algebra
course, which was invented after number and function were well
understood.
