dx/dt =1, x(0,s)=0, dy/dt=x, y(0,s) = s, du/dt=(y-1/2x^2)^2, u(0,s)=e^s
I did well at the beginning to get x(t,s) =t and y(t,s)=1/2t^2 + s, but got stuck with the du/dt part.
You can sub in x=t and y=1/2t^2 +s for x and y to get du/dt = s^2. But that's still three variables, and I can't see...
Ohhh! Yes, of course. If the second ball remained at rest and the first ball went forward, that would make no sense! And after the second collision the signs reverse for the non-zero velocity, so that would also be a contradiction.
Thanks for your reply and that does make sense. Though if they aren't equal, I get a similar equation with two solutions (one equal to zero and one non-zero), but in that case the velocities wouldn't have a clean exchange.
The simplest version of non-equal masses would be if the first ball...
I was thinking about ballistic pendulums and the symmetry they exhibit. In the simplest case, you have one ball that begins at a certain height and collides with another ball at rest. You can calculate via conservation of momentum and energy the new velocities and max vertical displacements...
I think you'd be right if my vector, x, was (x, y, z), but it's (x, y, 1). I was doing that to account for 2bx and a 2dy terms (whereas the 2 x 2 only has x^2, y^2, and xy terms). Perhaps this step isn't valid, but without my choice of this vector, I don't know how you'd model something like...
While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for xT A x, where x...
I seem to be getting an unsolvable integral here (integral calculator says it's an Ei function, which I've never seen). My thought was to use Bernoulli to make it linear and then integrating factors. Is that wrong? The basic idea is below:
P(x) 1, Q(x) = 1/2(1-1/x), n=-1, so use v=y^1-...
Hmm, yeah, maybe it was just the wording that is a little ambiguous. It says for a=2 AND b≠-1, the solution is inconsistent (makes sense, because then you get something like 0=1), for a≠2 AND b+4a^2-4a-7≠0, it is unique, and for a=2 AND b=-1 there are infinitely many solutions (also makes...
It makes sense that a=2 would cause problems because then we wouldn't have a matrix of full rank and we'd be unable to determine a value for w. But the key also says that when b+4a^2-4a-7≠0. Why is that an issue? For example, if a=1, that just says implies that w=0. Through back-subsitution...
But isn't y_2 showing a length? And why would they spend so much time going through Simpson's if that isn't applied here? It seems fishy they would give f''''(x), point to the idea that the two estimates are the same, and ask for "best estimate." Plus since the average of Simpson = the...
I'm confident in calculating the beginning values. Only question I had was whether in this graph y2 is assumed positive or negative. Obviously f(pi)= negative value, but is y2 negative or is it = |f(pi)|. Assuming the latter, it all makes sense and when I sum the two Simpson's Rule estimates...