# Deriving the first-order system for this governing equation

currently
Homework Statement:
Derive first-order system for governing equation of this damped nonlinear vibration; then find and write down the corresponding critical point and Jacobian Matrix.
Relevant Equations:
mx'' = -8x' - x + 9x^3
I tried finding the solution of the equation itself but it hasn't helped! Links to concepts would be greatly appreciated...thank you...

Homework Helper
Hi,

The term 'governing equation' is slightly misleading in the exercise statement -- I think
mx'' = -8x' - x + 9x^3 is the governing equation.

You sure the composer doesn't mean 'characteristic equation' ?

Deriving a first order system is as simple as defining u = x' and you get a system of two first-order differential equations.

currently
Homework Helper
Homework Statement:: Derive first-order system for governing equation of this damped nonlinear vibration; then find and write down the corresponding critical point and Jacobian Matrix.
Homework Equations:: mx'' = -8x' - x + 9x^3

I tried finding the solution of the equation itself but it hasn't helped! Links to concepts would be greatly appreciated...thank you...
As @BvU said you can make two first-order equation from the given one by defining an other function y=x'. Then x'' = y'.
The problem does not want you to solve this non-linear system of equations. As I understand, find the critical points (X,Y) (fixed points, equilibrium points) where both x' and y' are zero. What are these points?
Find the Jacobian matrices at the fixed points, and find the eigenvalues.
From the eigenvalues, you get information about the character of solution near the fixed points.
Lissen to

Last edited:
currently
Homework Helper
Gold Member
Is that equation really non-linear? I don't think so.

As long as the dependent variable and all its derivatives appear to 1st order only, the equation is linear.
Other than that all I can say is "try Wolfram Alpha"!

Homework Helper
Is that equation really non-linear? I don't think so.

As long as the dependent variable and all its derivatives appear to 1st order only, the equation is linear.
Other than that all I can say is "try Wolfram Alpha"!
The original equation was mx'' = -8x' - x + 9x^3 where x(t) was the dependent variable - a nonlinear second order one. That equation should be replaced by a first-order system of equations, and find the critical point(s) and Jacobian matrix. The system of equation is linearized around the critical point, with coefficients equal to the elements of the Jacobian matrix. From the eigenvalues of the Jacobian at the critical point, one gets information about the kind and stability of the solution near that point.

currently and BvU
currently
Hi,

The term 'governing equation' is slightly misleading in the exercise statement -- I think
mx'' = -8x' - x + 9x^3 is the governing equation.

You sure the composer doesn't mean 'characteristic equation' ?

Deriving a first order system is as simple as defining u = x' and you get a system of two first-order differential equations.
Yes, that is the governing equation. Sorry, should have been 'a damped nonlinear vibration' instead of 'this damped nonlinear vibration' and with the equation written below.

BvU
Homework Helper
Gold Member
The original equation was mx'' = -8x' - x + 9x^3 where x(t) was the dependent variable - a nonlinear second order one. That equation should be replaced by a first-order system of equations, and find the critical point(s) and Jacobian matrix. The system of equation is linearized around the critical point, with coefficients equal to the elements of the Jacobian matrix. From the eigenvalues of the Jacobian at the critical point, one gets information about the kind and stability of the solution near that point.
Yes, I goofed. I was thinking y(x), not x(t). It is nonlinear.