Direction of oscillations (2nd order ODE)

In summary: If the solutions are moving away from the origin it's a sink and if they are moving towards the origin it's a source.
  • #1
110
1

Homework Statement


[tex] \frac{d\vec{Y}}{dt} =
\begin{bmatrix}
0 & 2\\
-2 &-1
\end{bmatrix}\vec{Y}
[/tex]
With an initial condition of [tex] \vec{Y_0} = (-1,1)[/tex][/B]

a) Find the eigenvalues
b) Determine if the origin is a spiral sink, source, or center
c) Determine the natural period and frequency of the oscillations
d) Determine the direction of oscillations in the phase plane
e) Graph it

Homework Equations

The Attempt at a Solution



I have the solutions, the eigens are [tex] \frac{-1}{2} \pm \frac{\sqrt{15}i}{2}[/tex]

Because λ is negative and the roots of the characteristic polynomial are complex the solutions oscillate and it is a sink.

The frequency and periods can be found with [tex] f = \frac{2π}{μ}[/tex]

But the directions of oscillation is where I have a problem. Again I have the answer to this problem (graphing it with mathematica gives the directions), but my question is how do I typically find the direction of oscillations for equations of this form using algebra or some other relationship that exists?

Edit: I remember there being something about u + iw vectors, but it's really fuzzy that was the beginning of the semester.
 
Last edited:
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  • #2
rmiller70015 said:

Homework Statement


[tex] \frac{d\vec{Y}}{dt} =
\begin{bmatrix}
0 & 2\\
-2 &-1
\end{bmatrix}
[/tex]
With an initial condition of [tex] \vec{Y_0} = (-1,1)[/tex][/B]

a) Find the eigenvalues
b) Determine if the origin is a spiral sink, source, or center
c) Determine the natural period and frequency of the oscillations
d) Determine the direction of oscillations in the phase plane
e) Graph it

Homework Equations

The Attempt at a Solution



I have the solutions, the eigens are [tex] \frac{-1}{2} \pm \frac{\sqrt{15}i}{2}[/tex]

Because λ is negative and the roots of the characteristic polynomial are complex the solutions oscillate and it is a sink.

The frequency and periods can be found with [tex] f = \frac{2π}{μ}[/tex]

But the directions of oscillation is where I have a problem. Again I have the answer to this problem (graphing it with mathematica gives the directions), but my question is how do I typically find the direction of oscillations for equations of this form using algebra or some other relationship that exists?

Edit: I remember there being something about u + iw vectors, but it's really fuzzy that was the beginning of the semester.

Your DE makes no sense: it equates a vector on one side to a matrix on the other side. Did you mean
[tex] \frac{d}{dt} \vec{Y}(t) = \begin{bmatrix}
0 & 2\\
-2 &-1
\end{bmatrix} \vec{Y}(t) \: ?
[/tex]
 
  • #3
Ray Vickson said:
Your DE makes no sense: it equates a vector on one side to a matrix on the other side. Did you mean
[tex] \frac{d}{dt} \vec{Y}(t) = \begin{bmatrix}
0 & 2\\
-2 &-1
\end{bmatrix} \vec{Y}(t) \: ?
[/tex]
Yes I did, sorry about that forgot my second vector.
 
  • #4
So, I figure it out eventually and just in case any futuristic internet people find this I figure I can be helpful by saying if you use the point (1,0) and plug it into the original equation and set it equal to the initial condition vector you can draw an arc from the point (1,0) to the next point and consider if the solution is a source or sink.
 

1. What is the direction of oscillations in a 2nd order ODE?

The direction of oscillations in a 2nd order ODE (ordinary differential equation) depends on the coefficients of the equation. Specifically, it is determined by the sign of the coefficient of the second derivative term. If the coefficient is positive, the oscillations will be in the positive direction, and if the coefficient is negative, the oscillations will be in the negative direction.

2. How do the initial conditions affect the direction of oscillations?

The initial conditions, such as the initial position and velocity, also play a role in determining the direction of oscillations in a 2nd order ODE. If the initial conditions are such that the system is initially moving in the positive direction, the oscillations will continue in that direction. Similarly, if the initial conditions result in an initial negative direction, the oscillations will continue in the negative direction.

3. Can the direction of oscillations change during the course of solving a 2nd order ODE?

Yes, the direction of oscillations can change during the course of solving a 2nd order ODE. This can happen when the coefficient of the second derivative term changes sign, or when the initial conditions change. It is important to keep track of these changes in order to accurately model the system.

4. How do the parameters in the equation affect the direction of oscillations?

The parameters in the 2nd order ODE, such as the mass and spring constant, can also affect the direction of oscillations. These parameters can change the frequency and amplitude of the oscillations, which in turn can impact the direction of the oscillations. It is important to consider these parameters when analyzing and solving 2nd order ODEs.

5. Can the direction of oscillations be determined without solving the 2nd order ODE?

In some cases, the direction of oscillations can be determined without actually solving the 2nd order ODE. For example, if the coefficient of the second derivative term is known to be positive, it can be determined that the oscillations will be in the positive direction. However, in most cases, solving the equation is necessary to accurately determine the direction of oscillations.

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