System of diff equations

  • Thread starter Tarhead
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I am having problems with solving systems of differential equations.

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

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