Finding the optimal solution

Question

Find the optimal solution of the problem $$\min \Bigg \{ \int_0^1 [x^\prime (t)^2 + 2x(t)^2]e^t dt : x(0) = 0, x(1) = e - e^{-2}\Bigg \}$$ and the value of the minimum.

Not sure how to approach this question.

Answer 1

Write $F(x', x, t) = (x'^2 + x^2)e^t$. Then the integral of $F$ attains an extreme if

$$\frac{d\ }{dt} \frac{\partial F}{\partial x'} - \frac{\partial F}{\partial x} = 0$$

This will give you an ODE in $x$, which is straight forward to solve. Use that $x$ to evaluate the integral.