Second Order Ode with linearization I am given the following non-linear second order ode: 
$$x''=mg\sin(\theta)+k(L-\sqrt{x^2+h^2})\left(\frac{x}{\sqrt{x^2+h^2}}\right)-bx'$$
where $x$ is a function of $t$ or time. I wrote this equation as the following system: 
$$x_1'=x_2$$
$$x_2'=mg\sin(\theta)+k(L-\sqrt{x_1^2+h^2})\left(\frac{x_1}{\sqrt{x_1^2+h^2}}\right)-bx_2$$ 
where $x_1=x$ and $x_2=x'.$ The equilibrium points of this system satisfy the equation $x_1'=x_2'=0$  which means that $$x_2=0$$ and $$mg\sin(\theta)=kx_1\left(1-\frac{L}{\sqrt{x_1^2+h^2}}\right).$$ If we denote the solutions to the latter equation as $x_1^*$ then $(x_1^*,0)$ are fixed points of our system. Then we need to check their stability and for that we perturb our system. In other words we do the following: $$u=x_1-x_2^*$$
$$v=x_2-0.$$ Then as $u'=x_1'=f(x_1^*+u,x_2)$ by the Taylor series expansion centred at the fixed point we obtain $$u'=u\frac{\partial f}{\partial x_1}+v\frac{\partial f}{\partial x_2}+\text{higher order terms},$$ where $x'=f(x_1,x_2)=x_2.$ So we have that $$u'=v.$$ And similarly $$v'=u\underbrace{\left(kL\left(\frac{1}{\sqrt{(x_1^2+h^2)}}-\frac{x_1^2}{(x_1^2+h^2)^{3/2}}\right)-k\right)}_{f(x_1)}-bv+\text{higher order terms}.$$
Hence we can get the following linearised system: 
$$\begin{bmatrix}
    u' \\
    v'
\end{bmatrix}=\begin{bmatrix}
    0 & 1\\
    f(x_1) & -b
\end{bmatrix}\begin{bmatrix}
    u \\
    v
\end{bmatrix}.$$
Have I made any mistake in my analysis till here because I am getting the wrong conclusions after this? Perhaps someone can give some suggestions?  
 A: First of all, with your analysis you have managed to transform your system's equilibrium to the origin, that is, $(0,0)$ to be the solution of the following system:
\begin{equation}u' = 0\\
v'=0\end{equation}
Now, in order to compute the linearized system that accords to your own, you have to compute the Taylor expansion locally and close to the fixed point, that is, the components:
\begin{equation} \frac{\partial}{\partial x_1}f_1 , \frac{\partial}{\partial x_2}f_2\end{equation}
should be computed at $(x_1,x_2)=(x_1^*,x_2^*)$. The linearization should lead to a linear system:
\begin{equation} u' = \alpha u + \beta v \\ v' = \gamma u + \delta v \end{equation}
which in vector form can be written as:
\begin{equation} \begin{bmatrix}{u'\\v'}\end{bmatrix}=\begin{bmatrix}\alpha & \beta \\ \gamma & \delta \end{bmatrix}\begin{bmatrix}u \\ v\end{bmatrix}\end{equation}
Notice that the associated matrix should be linear. In your study, you clearly have a nonlinear function $f(x_1)$ instead of a number. So, to answer more properly, I will give you a quick notion of how to linearize your system properly. Consider an $n-$dimensional vector field:
\begin{equation}X'=G(X) \end{equation}
where $X=(x_1,x_2,...,x_n)$ is an $n-$ dimensional vector. $G=(g_1(X),g_2(X),...,g_n(X))$ is the vector of the right hand side functions. In your case, n=2 and 
\begin{equation} X = \begin{bmatrix} u \\ v \end{bmatrix} , G = \begin{bmatrix}u' \\ v' \end{bmatrix}\end{equation}
To compute the associated linearized system at a fixed point (say, the origin $(0,0)$), you have to evaluate the following matrix:
\begin{equation} J=\begin{bmatrix}\frac{\partial}{\partial x_1} g_1 & \frac{\partial}{\partial x_2}g_1 & \cdots & \frac{\partial}{\partial x_n}g_1  \\\frac{\partial}{\partial x_1}g_2 & \frac{\partial}{\partial x_2}g_2 &  \cdots & \frac{\partial}{\partial x_n}g_n \\\vdots & \vdots & \ddots & \vdots \\\frac{\partial}{\partial x_1}g_n & \frac{\partial}{\partial x_2}g_n &\cdots & \frac{\partial}{\partial x_n}g_n    \end{bmatrix} \end{equation}
The above matrix is called the Jacobian matrix. Remember that every derivative is computed at the fixed point i.e. at the origin where $(u,v)=(0,0)$. The Jacobian matrix is the coefficient matrix of the linearized system: 
\begin{equation} X'=JX \end{equation}
and for the 2-dimensional case:
\begin{equation} \begin{bmatrix} u' \\ v' \end{bmatrix} = J \begin{bmatrix} u \\ v \end{bmatrix} \end{equation}
I hope I have helped you somehow. A good introduction to linear algebra and dynamical systems is the following aticle and the references therein: 
Scholarpedia-Equilibrium
