In order to find streamlines for vector field, we need to solve this system of differential equations:(adsbygoogle = window.adsbygoogle || []).push({});

[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

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# Homework Help: Solution to system of differential equations

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