Recent content by mathador

Any pointers and/or help with proving the following would be appreciated For every prime number n, there exist positive integers (k,l,m) such that k(m2-n2)=2(m3+n3-lm) some examples (there could be several/many solutions for a given n) {n,k,l,m} {2, 14, 8, 2} {3, 59, 264, 1} {5...

Hi Kotreny, It's a nice observation. The geometric meaning is illustrated in the figure (I modified the figure found at http://mathworld.wolfram.com/PythagoreanMeans.html). Mathador

well, formally one can divide the two equations dx/dy=x/y, so dx/x=dy/y, yielding ln(abs(x/y))=const. HOWEVER, a first integral should be a continuous function, and the above isn't. Intuitively, your equations describe a source at x=y=0, and therefore phase volumes grow in time (i.e...

Hi, The trick is to define a new variable, y'=dy/dx=p(y(x)). Now y''=dp/dy*y'=dp/dy*p=f(y). Integrating now yields p^2=c+int(f(y))dy. The next step is to integrate dx=dy/p(y). Mathador

Hi Coto, Your example is an autonomous system (does not depend on time explicitly), therefore the dimension of its phase space is 2 (i.e. there is no extended phase space for this problem). When time enters the problem explicitly (for example, as in a forced system), the non-autonomous...

Hi, Try to express where the four lines x=0, x=2, y=0, y=2 (l1,..,l4) and their four intersections (p1,..,p4) get mapped under your map (M). The four new points (M(p1),...,M(p4)) determine a trapezoid, say T1. You can calculate the slopes of the lines connecting M(p1),...,M(p4). The images of...