Write and solve the Euler equations to make the following integrals stationary. \(\int_{x_{1}}^{x_{2}} \sqrt{x} \sqrt{1+y^{2}} d x\)

The Euler-Lagrange equation is a fundamental equation in the calculus of variations. It helps find functions that extremize (maximize or minimize) functionals. In simpler terms, it is used to determine the path, curve, surface, etc., for which a given integral is stationary (not changing).
This equation is derived from the principle of stationary action. The general form of the Euler-Lagrange equation is:
\[ \frac{d}{dx} \left( \frac{\partial L}{\partial \dot{y}} \right) - \frac{\partial L}{\partial y} = 0 \]
Here, \(L\) is the integrand (also known as the Lagrangian) and depends on \(x\), \(y\), and \(\dot{y} = \frac{dy}{dx}\). The solutions to the Euler-Lagrange equation provide the extremal functions required to make the integral stationary.

A functional is a map from a space of functions to the real numbers. Think of it as a function, but instead of taking numbers as its inputs, it takes functions as inputs.
In the context of the given problem, the integral we wish to make stationary is a functional:
\[ \int_{x_{1}}^{x_{2}} \sqrt{x} \sqrt{1+y^{2}} \, dx \]
The integrand \(\sqrt{x} \sqrt{1+y^{2}}\) is dependent on both \(x\) and \(y\). Our goal is to find a function \(y = y(x)\) that makes this functional stationary using the Euler-Lagrange equation. The functional tells us how much 'action' is associated with a certain function. The extremal function is what minimizes or maximizes this action.

A differential equation relates a function to its derivatives. Solving a differential equation means finding a function that satisfies this relationship.
In the last step of solving the Euler-Lagrange equation, we end up with a differential equation. This is because the resulting equation after applying the Euler-Lagrange equation involves derivatives of the function we are trying to find.
For the given problem, after substituting the functional into the Euler-Lagrange equation and simplifying, we obtain a differential equation which needs to be solved to find \(y(x)\), the function that makes the integral stationary. Solving this differential equation is the last critical step in finding the function that optimizes the given integral.