# Linearization of Second Order Differential Equations

by Ruby Tyra
Tags: linearization, pendulum
 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 $\theta_{1}''$$\normalsize = function$ [$\theta_{1}$,$\theta_{2}$,$\theta_{1}'$,$\theta_{2}'$] (The original equation is found at http://www.myphysicslab.com/dbl_pendulum.html, the equations inside the orange rectangle.) I was told to replace that function by a linear function of all four variables but I don't know where to start with that since the original equation is much more complex than the simple pendulum example we were given. Thank you!
 Quote by Ruby Tyra I'm having some difficulties figuring out how to linearize second order differential equations for a double pendulum. I have an equation that is in the form of $\theta_{1}''$$\normalsize = function$ [$\theta_{1}$,$\theta_{2}$,$\theta_{1}'$,$\theta_{2}'$] (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!
$$f(x,y,z,w) = f(x_0,y_0,z_0,w_0)+\frac{\partial{f}}{\partial{x}}(x-x_0)+\frac{\partial{f}}{\partial{y}}(y-y_0)+\frac{\partial{f}}{\partial{z}}(z-z_0)+\frac{\partial{f}}{\partial{w}}(w-w_0)$$
where the partials are evaluated at $$x_0,y_0,z_0,w_0$$