Register to reply 
Linearization of Second Order Differential Equations 
Share this thread: 
#1
Jan213, 02:48 PM

P: 1

I'm having some difficulties figuring out how to linearize second order differential equations for a double pendulum.
I have an equation that is in the form of [itex]\theta_{1}''[/itex][itex]\normalsize = function[/itex] [[itex]\theta_{1}[/itex],[itex]\theta_{2}[/itex],[itex]\theta_{1}'[/itex],[itex]\theta_{2}'[/itex]] (The original equation is found at http://www.myphysicslab.com/dbl_pendulum.html, the equations inside the orange rectangle.) I was told to replace that function by a linear function of all four variables but I don't know where to start with that since the original equation is much more complex than the simple pendulum example we were given. Thank you! 


#2
Jan213, 08:33 PM

Sci Advisor
HW Helper
Thanks
PF Gold
P: 5,059

[tex]f(x,y,z,w) = f(x_0,y_0,z_0,w_0)+\frac{\partial{f}}{\partial{x}}(xx_0)+\frac{\partial{f}}{\partial{y}}(yy_0)+\frac{\partial{f}}{\partial{z}}(zz_0)+\frac{\partial{f}}{\partial{w}}(ww_0)[/tex] where the partials are evaluated at [tex]x_0,y_0,z_0,w_0[/tex] 


#3
Jan313, 05:54 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,345

Note that what Chestermiller is saying is essentially the same as replacing the function by a Taylor polynomial in all variables, then dropping all but the linear terms. And that, in turn, is the same as replacing the "surface" by its "tangent plane".



Register to reply 
Related Discussions  
First order differential equations.  Calculus & Beyond Homework  3  
United States Elementary Differential Equations  1st Order Differential Equations  Calculus & Beyond Homework  1  
2nd Order RungeKutta: 2nd Order Coupled Differential Equations  Calculus & Beyond Homework  3  
Second order linear differential equations nonhomogeneous equations  Calculus & Beyond Homework  1  
Second order differential equations  Educators & Teaching  0 