CoV5

Constaint problem: Minimize $T(y)=\int_0^1\left(y'^2+x^2\right)\,dx\,$ s.t. $K(y)=\int_0^1y^2\,dx=2\,$ .

Using the method of Langrange multipliers,

$\int_0^1\left(y'^2+x^2+\lambda y^2\right)\,dx = J(y)\,$

If the Euler-Lagrange equations are satisfied then this integral has an extremum.

$\frac{\partial}{\partial y}\left[y'^2+x^2+\lambda y^2\right]=\frac{d}{dx}\frac{\partial}{\partial y'}\left[y'^2+x^2+\lambda y^2\right]\,$

$2\lambda y = 2y''\,$

$y''=\lambda y\,$ so $y(x) = c_1\cos\sqrt{\lambda}x + c_2\sin\sqrt{\lambda}x\,$

$y(0)=0\,$ gives $c_1=0\,$

$y(x)=c_2\sin\sqrt{\lambda}x\,$

$y(1)=0\,$ gives $0=c_2\sin\sqrt{\lambda}\,$ so $\sqrt{\lambda}=n\pi, n=1,2,...\,$

The solution is

$y(x)=c_2 \sin n\pi x\,$

$2=\int_0^1y^2\,dx = c_2^2\int_0^1\sin^2(n\pi x)\,dx\,$

$2=\frac{c_2^2}{2}\int_0^1(1-\cos 2n\pi x)\,dx = \frac{c_2^2}{2}\,$

$c_2=\pm 2\,$