How Do You Sketch Phase Plane Diagrams for Differential Equations?

[EV33]

Homework Statement

In general, how do you draw the phase plane for the given solution?

x=C1*e^{lambda1*t}[a1 a2]^{T} + C2*e^{lambda2*t}[b1 b2]^{T}

I think I know how to get the four asymptotic lines.

I am not sure how to determine the direction of my asymptotic lines
or how to draw the other solutions.

Homework Equations

C1,C2,a1,a2,b1,b2 are constants
T means transpose
t is the variable

The Attempt at a Solution

1.) So we have to draw four lines.

Let c1>0,c2=0
c1<0, c2=0
c2>0,c1=0
c2<0, c1=0

Each one of these will give you functions of x^{(1)} and x^{(2)}.
You will also be able to determine that x^{(1)} or x^{(2)} is less than/greater than 0 based off of the innequalities.

2.) I have no idea on how to determine the direction of the asymptotic lines.

3.) Also I don't know where to start my the solutions and where to end them when I draw them.



Thank you for your time.

 

[EV33]

ok I just figured out how to get the directions of the asymptotic lines so that just leaves my last question.

How do you determine the path of the various solutions?