A Calculus please help

Question

This is kinda a typical question of. But I'm really stuck in this.

Is there any one how to solve thisquestion. Really really appreciate it.

Answer 1

The optimal $y$ in the functional $$\int_a^b L(y,y',x)\, dx$$ must satisfy the Euler-Lagrange equation: $$\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'} = 0$$ Here, we have $L(y,y',x) = (y')^2+12xy$. Plugging this into the ELE gives us $$12x - \frac{d}{dx}(2y') = 12x - 2y'' = 0$$ i.e. $$y'' = 6x$$ and so $$y = x^3+c_1 x+c_2$$ for some constants $c_1, c_2$ which you can solve for using $y(0) = 0, y(1)=1$.