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>The Euler-lagrangian equation gives the equations of motion that once solved give you a family of solutions that minimize the action. A unique solution is given by specifying boundary conditions. It is just a case of inputing those boundary conditions. 
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>Wlog let $ x(0)=0 $ and $x(t_0)=a $. Integrating $\ddot{x} = \frac{F}{m}$ gives the general solution $x(t)=\frac{F}{2m}t^2 +Bt + A$, fixing C. Subbing in $x(0)=0$ gives $A=0$ and subbing $x(t_0)=a$ gives $B$ as $B=\frac{a - Ct_0^2}{t_0}$. ~ https://physics.stackexchange.com/a/664865/313317