Direction Fields for Systems of Differential Equations

[daveyman]

Homework Statement

Draw a direction field for the following differential equation:
$$x=c_1\left( \begin{array}{c} 1 \\ 1 \end{array} \right)e^t+c_2\left( \begin{array}{c} 1 \\ 3 \end{array} \right)e^{-t}$$

Homework Equations

N/A

The Attempt at a Solution

With a single differential equation, all you do is choose arbitrary x and y values and plug them into the differential equation. What you get is a slope, so you draw a little arrow with this slope at the point (x,y).

However, I don't know how to deal with a system of equations. If I choose arbitrary values for x and t, then I get multiple slopes. I can only draw one arrow at the point (x,t), so what do I do with multiple slopes?

Thank you in advance for your help!

 

[Unco]

Hi Daveyman,

That looks more like a solution to a system of differential equations to me. In which case, we can sketch the direction field by first drawing the two eigenvectors (1,1) and (1,3), and then, since the first corresponds to a positive eigenvalue (the exponent of e is positive) and the latter corresponds to a negative eigenvector (negative exponent), we can sketch solution curves alongside the former that are directed away from the origin and solution curves alongside the latter directed towards the origin. In this case, the origin is called an (unstable) saddle point.

 

[daveyman]

Thanks for your reply! So, you are saying I simply draw the two vectors and then draw curves tending towards/away from those vectors. Sounds easy enough, but would this be considered a direction field? Usually direction fields have lots of little arrows all over the place :-)

 

[Unco]

Arrows are necessary to indicate a curve directed towards or away from the origin.

From your reply, however, I'm a bit concerned about the relevance of my response. If you are unfamiliar with the concepts I described, then that won't be what is required of you. Could you provide some context to the question? It would be conceivable to have it as an exercise in plotting solutions curves with software, for example: choosing initial conditions, solving for c_1 and c_2, then plotting parametric plots.

 

[daveyman]

Actually your response was very helpful. I understand now - thanks!