I have to sketch the phase flows for the following systems:
(a) $r'= (1-r^2)cosh(r)$ , $θ'=1$ ( $(r,θ)$ are polar coordinates).
(b) $x'=2xe^x$ , $y'=-y$
Considering the first equations of (a) and (b), I dont find a way to solve them explicitly (and I guess that maybe I dont have to, since the question is just "sketch the flows").
My question is, is there a programm where I can put these systems of equations and then give me the flows?
You can use Wolfram Alpha and the StreamPlot function:
https://www.wolframalpha.com/input/?i=StreamPlot%5B%7B2xe%5Ex%2C+-y%7D%2C+%7Bx%2C+-2%2C+2%7D%2C+%7By%2C+-3%2C+3%7D%5D
https://www.wolframalpha.com/input/?i=StreamPlot%5B%7B%281-r%5E2%29cosh%28r%29%2C1%2B+1e-15theta%7D%2C+%7Br%2C+-2%2C+2%7D%2C+%7Btheta%2C+-3%2C+3%7D%5D
(although probably it's quite easy to sketch them by hand by linearization, see this for example)