when solving to a problem I get a "trivial" solution: $$ \int_{0}^{1}tyy' dt $$ Subject to $$ y(0)=0\\y(1)=1 $$
When applying the Euler's equation: $$ F_{y'y'}y^{''}(t)+F_{yy'}y^{'}(t)+F_{ty'}-F_{y}=0 $$ The partial derivatives being: $$ F_{y'y'}=0\\F_{yy'}=t\\F_{ty'}=y\\F_{y}=ty^{'} $$ Applying euler's equation I get: $$ y(t)=0 $$ Which is a contradiction since $$y(1)=1$$ Appreciate your help.
NOTE: I got this exercise from "Elements of Dynamic Optimization" Chiang, Alpha.
This is Ex. 7 on p. 36, which (tellingly) asks to find the extremals, if any. The textbook clearly illustrates the source of the problem on pp. 35-36.
More specifically, when $F$ is linear in $y^\prime$, then $F_{y^\prime}$ is a constant and $F_{y^\prime y^\prime} = 0$. So the Euler equation no longer yields a second-order differential equation, and then it is not possible in general to adapt the time path to the boundary conditions. Unless the path happens to fit the boundary condition, there is no extremal.