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Test macros for x1, x2, y1, y2 whereas these should not be changed: x_1, x_2, y_1, y_2. Here is an inline version: y2 = y1 + f1(x1) + f2(x2) of a math equation. And again these should not be changed: x_1, x_2, y_1, y_2.

Here is an example of using the eqn_start macro:

y2 = y1 + f1(x1) + f2(x2)

Here is a MathJax version \(y_1 + f1(x_1) + f2(x_2)\) which is inline within text.

The Euler-Lagrange Equation

The Euler-Lagrange equation is a differential equation that \(y(x)\) must satisfy in order to minimize (or possibly maximize) an integral of the form $$ I = \int_{x_1}^{x_2} f(x, y, y') dx $$ where also \(y(x)\) must pass through the two fixed end points \(y(x_1) = y_1\) and \(y(x_2) = y_2\). Through reasoning about families of curves that include the minimizing curve \(y(x)\), we can get the Euler-Lagrange equation: $$ \frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) = 0 $$

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