Second Order DE: Nonlinear Homogeneous

In summary: Next, if you are lucky enough to know the value of the second derivative of u at x, you can do a "6-point" Taylor series expansion about x and arrive at an exact solution.
  • #1
irighti
1
0
I am sure most of you are familiar with the equation: m(x)''+c(x)'+k(x) = 0. Then, we create an auxillary equation that looks like this: mr^2+cr+k = 0. And, then we find the roots of this auxillary equation, calling them r1 and r2. And, if the roots are r1,r2>0 we consider the system to be overdamped and we develop the following general form of the equation to be: x=A*e^(r1*t)+B*e^(r2*t) etc...

I have an inverted pendulum with damping and understand the equation to be in the form of a second order nonlinear (homogenous) differential equation.

The second order nonlinear homegenous differential equation is: m(x)''+c(x)'-k*sin(x)=0.
I have tried everything I know and haven't had much luck... I mean how do you find the roots to something like this? I am sure there is a method to this madness. I would think there is some type of variation of parameters or substitution involved but do not know how to apply substitution etc... to this application.

I would appreciate any help or advice that might lead me in the right direction...

Thanks in advance, John
 
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  • #2
There are no generally effective methods to gain the exact solution(s) of non-linear equations, differential or algebraic!

The "method to this madness" in order to get approximate solutions is to make a guess, giving you a wrong answer, then make a second guess, giving you (hopefully!) a less wrong answer, and so on ad infinitum.

Such systematized guess-working techniques are very much in use, and to improve those techniques is one of the main branches of applied mathematics areas like computational fluid dynamics and suchlike fields.
 
  • #3
Here, for example, since sin(x)= x- (1/6)x3+ ..., a first approximation would be to replace sin(x) by x: mx"+ cx'- kx= 0.

A slightly more sophisticated method is "quadrature": Let u= x' so that x"= u'. By the chain rule, u'= (du/dx)(dx/dt)= u u' so the equation becomes u u'+ cu- kx= 0. That is now a first order equation for u as a function of x. The problem typically is that even after you have found u, integrating x'= u, to find x as a function of t, in closed form may be impossible (without the damping, this gives elliptic integrals).
 

What is a second-order nonlinear homogeneous differential equation?

A second-order nonlinear homogeneous differential equation is an equation that involves a second derivative of a dependent variable and contains nonlinear terms, meaning the dependent variable is raised to a power other than 1. Homogeneous means that the equation is equal to zero and does not contain any constant terms.

How is a second-order nonlinear homogeneous differential equation solved?

The general method for solving a second-order nonlinear homogeneous differential equation is by using substitution and integration. The substitution involves replacing the dependent variable with a new variable, which transforms the equation into a separable equation that can be solved by integrating both sides.

What is the role of initial conditions in solving a second-order nonlinear homogeneous differential equation?

The initial conditions are necessary in solving a second-order nonlinear homogeneous differential equation because they provide specific values for the dependent variable and its derivatives at a given point. These conditions are used to determine the arbitrary constants that arise during the integration process and make the solution unique.

What are some real-life applications of second-order nonlinear homogeneous differential equations?

Second-order nonlinear homogeneous differential equations have many real-life applications in various fields of science, such as physics, chemistry, and biology. They can be used to model physical phenomena like pendulum motion, chemical reactions, and population growth. They are also essential in engineering and control systems, such as in designing circuits and predicting the behavior of mechanical systems.

What are the challenges in solving second-order nonlinear homogeneous differential equations?

One of the main challenges in solving second-order nonlinear homogeneous differential equations is that they do not have a general formula or algorithm for finding solutions. Each equation requires a different approach and may involve difficult integration or the use of advanced mathematical techniques. Additionally, determining the initial conditions accurately can be challenging, and small changes in these values can significantly affect the solution.

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