Thread: Draw a phase protrait?

I was given an problem where i had to find the general solution then draw a phase portrait, then classify the origin and determine the stability of the system. I solved for the general solution but i have no idea how to do the other parts

general solution is

x=c_1(1,1)e^2t+c_2*(1,-4)*e^-3t (1,1 and 1,-4 are both 1 column 2 row matrices, i am not sure how to create a matrix on this forum)

for a 2nd order DE
find critical pts from the DE... i.e. where dy/dx = 0 or undefined
draw a vertical line representing the y-axis, this will be the phase line
indicate the positions of the critical points (constant/equilibrium solutions, so draw horizontal lines at those points) on the phase line
notice this will divide the line into regions
use (d^2)y/d(x^2) to determine the sign of the solution y(x) in those regions
put an ^ symbol on the phase line if the sign of that region determined from the 2nd deriv is positive, and an upsidedown ^ if the sign of that region is negative

the dots represent your critical points on the line, tilt your head sideways > . < asymptotically stable, < . < or > . > semistable, < . > asymptotically unstable (referring to the stability of that point)

the stability of these points may also be identified by classifying those points as attractors (stable) or repellers (unstable)