I'm reading about second-order odes, and one of the chapters is about the Sturm-Liovuile method. The chapter 10, from Arfken, begins by talking and defining $L$ to be:
$$Lu(x) = p_0(x)\frac{d^2}{dx^2}u(x)+ p_1(x)\frac{d}{dx}u(x) + p_2(x)u(x)$$
Then it says:
For a linear operator $L$, the analog of a quadratic form for a matrix in Chapter $3$ is the integral
$$\langle u|L|u\rangle = \langle u|Lu\rangle = \int_a^b u(x)Lu(x)\ dx =$$ $$\int_a^b u\{p_0u''+p_1u'+p_2u\}\ dx$$
By integrating by parts once or twice, we are led to the equivalent expression:
$$\langle u| L| u\rangle = [u(x)(p_1-p_0')u(x)]_{x=a}^{b} + \int_a^b \left\{\frac{d^2}{dx^2}[p_0u]-\frac{d}{dx}[p_1u]+p_2u \right\}\ u \ dx\tag{10.3}$$
Because of the analogy with the transposed matrix in Chapter 3, it is convenient to define the linear operator in Eq. 10.3,
$$\overline{L}u = \frac{d^2}{dx^2}[p_0 u]-\frac{d}{dx}[p_1u]+ p_2u = $$ $$p_0\frac{d^2u}{dx^2}+(2p_0'-p_1)\frac{du}{dx}+(p_0''-p_1'+p_2)u$$
The Chapter 3 is all about matrices.
What is an 'analog of a quadratic form form a matrix'? Why is he doing things like this? Is he doing an equivalence between matrices and linear operators? Could somebody really tell me what's going on? Is he trying to solve the equation $Lu(X)$?
He is trying to motivate the discussion of self-adjoint differential operators by connecting it to what you already know about self-adjoint matrices. To see $\int u(x) Lu(x) \, dx$ is analogous to a quadratic form, think of $L$ as an $n\times n$ matrix, and $u$ as an $n$-dimensional vector. Then $\langle u, Lu \rangle = \sum_{x=1}^n u(x) \cdot (Lu)(x)$ where I have written $u(x)$ to be the $x$th component of $u$ and similarly for the vector $Lu$. Written in this way, you can see that you can get to the integral expression by replacing the summation with an integral. So in this sense, you can think of this operator as a "continuous" matrix. Hope this helps.