- #1
Alteran
- 18
- 0
In order to find streamlines for vector field, we need to solve this system of differential equations:
[tex]\frac{dx}{dt} = -x+y+z[/tex]
[tex]\frac{dy}{dt} = x-y+z[/tex]
[tex]\frac{dz}{dt} = x+y-z[/tex]
where
[tex]x(0) = a[/tex]
[tex]y(0) = b[/tex]
[tex]z(0) = c[/tex]
I have used Mathematica in order to find solutions for these equations and it calculated the following:
[tex]x = \frac{2a-b-c+ae^{3t}+be^{3t}+ce^{3t}}{3e^{2t}}[/tex]
[tex]y = \frac{-a+2b-c+ae^{3t}+be^{3t}+ce^{3t}}{3e^{2t}}[/tex]
[tex]z = \frac{-a-b+2c+ae^{3t}+be^{3t}+ce^{3t}}{3e^{2t}}[/tex]
it is, of course, right - I have check parametric plot and it is streamline, but I think how it came to that solution? Does anybody can explain how I can solve these equations (step-by-step would be very very helpful :roll: ) without using of Mathematica? I do not understand why there is [tex]3e^{2t}[/tex]?
Thank you
[tex]\frac{dx}{dt} = -x+y+z[/tex]
[tex]\frac{dy}{dt} = x-y+z[/tex]
[tex]\frac{dz}{dt} = x+y-z[/tex]
where
[tex]x(0) = a[/tex]
[tex]y(0) = b[/tex]
[tex]z(0) = c[/tex]
I have used Mathematica in order to find solutions for these equations and it calculated the following:
[tex]x = \frac{2a-b-c+ae^{3t}+be^{3t}+ce^{3t}}{3e^{2t}}[/tex]
[tex]y = \frac{-a+2b-c+ae^{3t}+be^{3t}+ce^{3t}}{3e^{2t}}[/tex]
[tex]z = \frac{-a-b+2c+ae^{3t}+be^{3t}+ce^{3t}}{3e^{2t}}[/tex]
it is, of course, right - I have check parametric plot and it is streamline, but I think how it came to that solution? Does anybody can explain how I can solve these equations (step-by-step would be very very helpful :roll: ) without using of Mathematica? I do not understand why there is [tex]3e^{2t}[/tex]?
Thank you