Originally Posted by

**QDsolarX** I need to solve the Euler equation to make the following integral stationary:

$\displaystyle \int ( y'^2+\sqrt{y}) dx$

so I happily assume $\displaystyle F = y'^2+\sqrt{y} $ and find the partial derivative F wrt y' and F wrt y, put both into the Euler equation:

$\displaystyle \frac{d}{dx}(2y') -\frac{1}{2\sqrt{y}} = 0$

Route 1: simple rearrangement yield:

$\displaystyle y''- \frac{1}{4}y^{-\frac{1}{2}} = 0$

Route 2: intergate both side by x

$\displaystyle \int \sqrt{y} dy' = \frac{x}{4}$