Minimize integral

Dec 2015
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166
Earth
If \(\displaystyle \int_{0}^{\pi } y^2 dx =1 \; , \; y(0)=y(\pi)=0\) . Minimize \(\displaystyle I=\int_{0}^{\pi} y'^2 dx\).
should I use partial integration and where ? I'm a beginner on these type of problems and the book gives not a single hint.
 
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Dec 2015
1,078
166
Earth
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\(\displaystyle y=\pm \sqrt{2/ \pi } \sin(x)\) ; \(\displaystyle \; y(0)=y(\pi) =0\) (correct).
\(\displaystyle \; \; 2/ \pi \int_{0}^{\pi} \sin^2 (x) dx =2 / \pi \cdot \pi /2 =1\) (correct).
\(\displaystyle \min I=2/ \pi \int_{0}^{\pi} \cos^2 (x) dx =1 \) (correct).
 
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SDK

Sep 2016
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\(\displaystyle y=\pm \sqrt{2/ \pi } \sin(x)\) ; \(\displaystyle \; y(0)=y(\pi) =0\) (correct).
\(\displaystyle \; \; 2/ \pi \int_{0}^{\pi} \sin^2 (x) dx =2 / \pi \cdot \pi /2 =1\) (correct).
\(\displaystyle \min I=2/ \pi \int_{0}^{\pi} \cos^2 (x) dx =1 \) (correct).
This is a calculus of variations problem. You should take the functional derivative of $I$ and set it to zero to get the corresponding Euler-Lagrange equations. Where did you get that $y = \pm \sqrt{2/\pi} \sin (x)$?
 
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