System of coupled second order differential equations.

In summary, the conversation is about finding the characteristic equation for a second order coupled system of differential equations. The method suggested is to set x and y to standard results and convert the system into four first order equations, then find the eigenvalues of a 4x4 matrix. The characteristic polynomial is then obtained by multiplying the coefficients of the differential operators.
  • #1
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Hey folks I'm looking for a way to find the characteristic equation for a second order coupled system of differential equations such as...

[tex]\ddot{x} + A\dot{y} + Bx = 0[/tex]

[tex]\ddot{y} + C\dot{x} + Dy = 0[/tex]

Where x and y are functions of time.

I know I can solve it by setting x and y to standard results (trig, exponential) but I'd like to know a method to solving this rather than plug and solve for coefficients.

Specifically I'd like to know how to find the characteristic equation for this. I've tried setting it to a first order system but I can't see it leading anywhere (or perhaps I just did it wrong...).

I don't want a full answer, just the name of a method or something like that.
 
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  • #2
Actually I think I've got it...

Setting [tex]x_1 = x[/tex], [tex]y_1 = y[/tex], then converting into four first order differential equations. Find eigenvalues of 4x4 matrix etc... Think that's it.
 
  • #3
It's not entirely clear what you did other than add a subscript to x and y, but I suspect you have the right approach.
 
  • #4
rewrite the system (I will use E for your D and D for differentiation)
(D^2+B)x+(AD)y=0
(CD)x+(D^2+E)y=0
then the characteristic polynomial is
(D^2+B)(D^2+E)-ACD^2=
D^4+(B+E-AC)D^2+BE
as expected.
 

1. What is a system of coupled second order differential equations?

A system of coupled second order differential equations is a set of equations that describes the relationship between multiple variables, where each variable is a function of both time and one or more other variables. These equations are called "coupled" because the variables are interdependent, meaning that the value of one variable affects the value of another. The "second order" part refers to the fact that the equations involve second derivatives, meaning they describe how the rate of change of a variable changes over time.

2. What are some real-world applications of systems of coupled second order differential equations?

Systems of coupled second order differential equations are commonly used in fields such as physics, engineering, and biology to model complex systems. Some examples include the motion of a pendulum, the behavior of electrical circuits, and the dynamics of chemical reactions.

3. How do you solve a system of coupled second order differential equations?

The solution to a system of coupled second order differential equations can be found by solving each individual equation, and then using the values of the variables to solve for the other variables in the system. This can be done analytically, using mathematical techniques such as substitution and elimination, or numerically, using computer algorithms.

4. What is the difference between a coupled system of second order differential equations and a system of first order differential equations?

The main difference between these two types of systems is the number of derivatives involved. A system of first order differential equations involves only first derivatives, while a coupled system of second order differential equations involves second derivatives. This means that the latter is more complex and can describe more complex relationships between variables.

5. Can a system of coupled second order differential equations have multiple solutions?

Yes, a system of coupled second order differential equations can have multiple solutions. This is because the equations are often nonlinear, meaning that the relationship between the variables is not a simple one. In some cases, multiple solutions may exist that satisfy the equations, or there may be a range of possible solutions depending on the initial conditions.

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