I'm having a course on Calculus of Variations and I'm doing my first homework problems. One of them is the following:
Determine the smooth extremum which satisfies the boundary conditions for:
$$\min \int_0^1xy^2+x^3y\;dx,\;\;\;y(0)=A, \;y(1)=B,$$
where $y=y(x)$.
Can someone guide me how to solve this problem correctly? What I tried is the following:
I know that in order for $y^*$ be the optimal solution for the functional, then it must satisfy the Euler-Lagrange equation:
$$\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right)=0,$$
where in my problem $F(x,y,y') = xy^2+x^3y$ and $y'=\frac{dy}{dx}$. By applying the E-L equation I get:
$$2xy+x^3=0,$$
from where I solve that $$y=-\frac{1}{2}x^2.$$
Where should I apply the boundary conditions now? Did I make a mistake somewhere or is this how I'm supposed to solve it? Thank you for your help =)
Or does the boundary conditions in this case mean that: $y(0)=A=0 $ and $y(1)=B=-\frac{1}{2}$?
P.S. here is my problem in its original form:
