How to Determine Stability Intervals and Periodicity in Differential Equations?

[Tarhead]

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?

 

[M98Ranger]

Solving systems of differential equations can be a challenging task, so it is understandable that you are having some difficulties. It is important to have a clear understanding of the concepts and techniques involved in order to successfully solve these problems.

For the first problem, you have correctly set up the system of equations and found the characteristic equation. To find the intervals for a stable focus and stable node, you will need to consider the different values of gamma. Remember that for a stable focus, you want the real part of the eigenvalues to be negative, and for a stable node, you want the real part of the eigenvalues to be zero. So, you can use the quadratic formula to solve for r and then substitute different values of gamma to determine the corresponding intervals.

For the second problem, you have also correctly set up the system of equations and found the characteristic equation. To factor the equation, you can use the quadratic formula or try to find two numbers that multiply to -36 and add up to 0. Once you have found the values of r, you can plug them back into the system of equations and solve for x1 and x2. Then, you can use the initial conditions to find the specific solution. To determine the period, you can look for a pattern in the solution or use the fact that the period is equal to 2π/ω, where ω is the imaginary part of the eigenvalues. To find when the point is closest to the equilibrium point 0, you can set the solution equal to 0 and solve for t. This will give you the moment(s) when the point is closest to 0.

It is always helpful to practice solving different types of systems of differential equations to improve your understanding and skills. You can also consult with your instructor or classmates for additional guidance and resources. Good luck!