- #1

Tarhead

- 7

- 0

x'= [(-3 ) (gamma)]x

...[ ( 6 ) ( 4 ) ]

I am supposed tofind the interval of values of gamma for a) stable focus and b) stable node.

I started by

[(-3-r) (gamma)][x1] = [0]

[( 6 ) (4-r ) ][x2]...[0]

det(A-rI) = (-3-r)(4-r)-6(gamma) = 0

= r^2-r-12-6(gamma)= 0

but I don't know where to go after this point to find these different intervals.

For another problem:

x'= [0 3]x

...[-12 0] with initial conditions x1(0)= 1, x2(0) = 2

show that the solution x(t) is periodic and determine its period. Additionally to find the moment(s) when the point x(t) is closest to the equilibrium point 0.

For this I have

[(-r ) (3)][x1] = [0]

[(-12) ( -r)][x2]...[0]

so r^2 + 36 = 0

how do I or do I factor this? and after I find my values of r and plug them back in, where do I go?