DE question, how can I use straight line trajectories to come up with

[jlatshaw]

Ok, so this is a differential equation question.
How can I use the eigenvectors/eigenvalues to find the formulas for straight line trajectories and from those formulas that I come up with, how can I alter them so as to start at any given point that I would like them to (like with starter data)?

I know that I can come up with the straight line trajectories starting at the tips of the eigenvectors using:

{x1[t_], y1[t_]} = eigenvector[1] E^(eigenvalue[1] t)
{x2[t_], y2[t_]} = eigenvector[2] E^(eigenvalue[2] t)

Using the above, then if I want my solution plots to obey certain starter data, say x[0] = A, y[0] = B,
What do I have to do to the above to make the by solution plots start at those values?
Any suggested reading or videos on the subject to get a good understanding of it?

Thanks for any help,
-James

 

[jlatshaw]

Ok, so I figured it out:

If you have a DE system that is linear, then if we know eigenvalue and eigenvectors of a coefficient matrix A, we know that trajectory plots starting at the ends of the eigenvectors will be:

{x1[t_], y1[t_]} = eigenvector[1] E^(eigenvalue[1] t)
{x2[t_], y2[t_]} = eigenvector[2] E^(eigenvalue[2] t)

These will go with the flow. However, if we would like to start at any point, then we can say:

{x[t],y[t]} = C1 {x1[t], y1[t]} + C2 {x2[t], y2[t]}
where C1 and C2 are chosen, based off of the coefficients of {x1[t], y1[t]} and {x2[t], y2[t]} such that the resulting coefficients will be where you want the trajectory to start.
This is all possible because the system is linear.