Questions from a math 231 student, with answers (these are good questions, and I thought the answers would have general interest.)

I have a few questions concerning the differential equations stuff we've been covering:

1. The incontinuities in the domain of a solution are caused primarily by undefined places in the original differential equation, right? Discontinuities in the final solution also play a role here too, correct?

That's right. There is a difference between linear and nonlinear equations here, which we'll discuss later. In the  linear case: y'=f(x)y+g(x), y(x_0)=y_0, the
interval of definition of the solution is the largest interval (containing x_0) in which both f(x) and g(x) are defined. In the nonlinear case (as in y'=y^2),
although the "coefficients" are defined everywhere, in most cases solutions will not be defined on the whole real line. Again, this will be discussed in detail later.

2. Undefined behavior appears in many differential equations if you rearrange them. Like in assignment 2 problem 2, if we rearrange the equation in a seperable form we can get a 1/(y^2 - 1) construct. This gives us y != -1 and 1 in our solution of the function x(y) because of divide by zeros. If we don't rearrange, y can equal 1 if x is -2, and the differential equation is still satisfied. It seems like this could be a real problem since domains are necessary for our solutions.

That is a very good point. The general rule is that, although rearrangement may be needed in order to solve the equation, in the end the domain of the
solution must take into account the equation in the form originally given.

3. For the (ii) part of the first three problems (find solutions not in the one-parameter families), I'm getting single points primarily by solving the equations such that the coefficients of dx and dy are zero. Is this what you are looking for?

For an equation given in the form Pdx+Qdy=0, in many cases the singular solution will have the form y(x)=Y for all x, where Y is such that P(x,Y)=0 for
all x. It is always a good idea to look for solutions y=const.

4. What does E(x, f(x)) = constant tell us past the fact that we have an equation that we can implicitly derive and find functions of which E(x, f(x)) would serve as the equation part of a solution? Is this just some way to make implicit solutions make sense, seeing as how it may be impossible to get y = f(x) and y' = f(x, y)? It seems that any one-parameter solution would be a conserved quantity (but does conserved quantity mean anything past one-parameter solution?).

Well ,the difference is whether you focus on the concept of "conserved quantity" (which has physical meaning and occurs in a broader class of DEs)
or "implicit solution". Formally, they are very closely related (but not exactly the same). If the equation y'=f(x,y) for y=y(x) has the function E(x,y) as a conserved quantity, then  E(x,y)=C is a one-parameter-family of implicit solutions *and* any solution satisfies this relation. On the other hand, a differential equation
may admit a one-parameter family of solutions given implicitly by F(x,y)=C (where F is defined in some region of the plane), and, in addition, other
solutions not of this form. When this happens, one would not say that F is a conserved quantity (See example 6.66 on p.53, discussed in class: the function y(x)=1 for all x between -1 and 1 does not have the implicit form (c))

5. For this homework we're turning in tomorrow, we can only turn in 4/9 problems you've assigned, right?

Right. You may turn in more (I'll look at them if there is time) but only the first four among those you turn in will count towards a grade (1 pt per problem)