The problem is to find the equation that minimises the following functional: $$ J[y] = \int_0^1 \frac{1}{2}(y')^2 +yy'+y'+y \ dx. $$ The endpoints are not specified.
So far I have calculated the solution of the Euler Lagrange equation to be $$ y(x) = C_1x+C_2+\frac{x^2}{2}, $$ I am unsure of how to proceed with no other information.