# 5.6: Euler’s Integral Equation


An integral form of the Euler differential equation can be written which is useful for cases when the function $$f$$ does not depend explicitly on the independent variable $$x$$, that is, when $$\frac{\partial f}{\partial x}=0.$$ Note that

$\frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} \frac{dy}{dx}+\frac{\partial f}{\partial y^{\prime }}\frac{dy^{\prime }}{dx}$

But

$\frac{d}{dx}\left( y^{\prime }\frac{\partial f}{\partial y^{\prime }}\right) =\frac{\partial f}{\partial y^{\prime }}\frac{dy^{\prime }}{dx}+y^{\prime } \frac{d}{dx}\frac{\partial f}{\partial y^{\prime }}$

Combining these two equations gives

$\frac{d}{dx}\left( y^{\prime }\frac{\partial f}{\partial y^{\prime }}\right) =\frac{df}{dx}-\frac{\partial f}{\partial x}-y^{\prime }\frac{\partial f}{ \partial y}+y^{\prime }\frac{d}{dx}\frac{\partial f}{\partial y^{\prime }}$

The last two terms can be rewritten as

$y^{\prime }\left( \frac{d}{dx}\frac{\partial f}{\partial y^{\prime }}-\frac{ \partial f}{\partial y}\right)$

which vanishes when the Euler equation is satisfied. Therefore the above equation simplifies to

$\frac{\partial f}{\partial x}-\frac{d}{dx}\left( f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}\right) =0 \label{5.24}$

This integral form of Euler’s equation is especially useful when $$\frac{\partial f}{\partial x}=0,$$ that is, when $$f$$ does not depend explicitly on the independent variable $$x$$. Then the first integral of Equation \ref{5.24} is a constant, i.e.

$f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}=\text{constant}$

This is Euler’s integral variational equation. Note that the shortest distance between two points, the minimum surface of rotation, and the brachistochrone, described earlier, all are examples where $$\frac{\partial f }{\partial x}=0$$ and thus the integral form of Euler’s equation is useful for solving these cases.

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