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