well i am teaching ode starting tomorrow. i expect people to have perfect ability to reproduce definitions in all math classes, because imperfect definitions are useless. one word omitted from the definition of a continuous function, or anything in amth really, makes it wrong usually. that is because in mathn the definitions are very precise and very economical. no word is there that does not matter.
as to ode, i always disliekd it as a student because it was so compoutationally oriented, turn the crank, get an answer. no thjeory to illuminate it, so I will include some theory myself.
the best book on ode i have found for understanding it is the one by arnol'd.
an ode is simply a field of vectors, one at every point of your surface, or of space, or of the line, or whatever is your ambient space. a solution is a particle flowing along your space and having excatly the right velocity vector at every point, namely the one specified by the given field.
its like a road with speed signs everywhere, a solution is a car driving along at exactly the right speed at all times.
then the compoutational, solution is a formula for the position of the car at all times. this is hard to get, since we know from calculus that the formula for an integral is often mroe complicated than the formula for the integrand. in fact in the whole world of functions, the integral of a known functiuon is usually an unknown function.
thus it is likely that in ode, even if the field of vectors is specified by known functions, i.e. those to which we have given names, the function giuving the solution, i.e. the position of the moving particle, will be given by some new function.
so in ode, unless we want to makew the unnatural restriction that we will only study problems with answers given by functions we already know, we must fgace up to the most difficult issue for begionning students: namely what is a function and how much dow e need to nkow aboput it to know it?
a student almost always thinks a function is something given by a familiar formula, but in ode this is not so for most equations. so he/she must accept that a function is given by some limiting proicess that yields a result that ahs no nkown name. this is tough.'
hence in ode one should be as flexible as possible about functions. the first step toward solving an ode should by just to be able to sketch a rough idea of the graph of the solution.
e.g. let all the vectors be of unit length and horizontal. what do you think the motion looks like having those vectors as velocity vectors?
now let all the vectors at each point be perpendicular to the radius from that point to the origin, and counterclockwise in direction. what would the motion be?
then ask yourself if you can really tell the difference between these vectors and ones that tilt ever so slightly in towards the origin, i.e. making angle 89 degrees with the radius. if this is the case, what would the motion be?
then in the final throes of the subject, put forward a few specific formulas for the vectors and ask whether the motion also has similar formulas.
almost the only cases where the answer is known and simple is where the formula for the vectors is just given by linear functions (with constant coefficients), such as one asked elsewhere on this forum recently:
y' = x+y.
try sketching the vectors for y' = -x and y = x'.
then try y = x' and y' = -sin(x).
I agree with ranger. Doing some problems(not easy ones ) let you know whether you've learned something or just memorized it!
You know some people think that if you understand something, you're able to reproduce it word per word.
