5. Euler's equation for the simplest problem.

Theorem 1: If a functiony  is extremal to the function

wherey belongs to C2[a,b] and

then y  must fulfill the Euler equation

Remark: The Euler equation corresponds to the condition

in extreme value problems and is an ordinary differential equation that in general is nonlinear. By using the chain rule, we see that it can be expanded as

The function L=L(x,y,y) is called the Lagrangian of the problem. In our previous problems problems 45 and 6  the Lagrangians were