Transform Dirichlet condition into mixed boundary condition

Question

If I have a homgeneous linear differential equation like this one (or any other eq): $$y''(x)-y'(x)=0$$ And they give me these Dirichlet boundary conditions: $$y(0)=y(1)=0$$ Can I transform them into a mixed boundary conditions?: $$y(0)=y'(1)=0$$ I tried solving the equation, derivating it and using the original boundary conditions but I don't get anything.

Answer 1

No, it is in general not possible to obtain conditions on the derivative of a function just from its value at a point.

Note that setting $u=y'$ you need to solve $u'=u$, then integrate $u$ to obtain $y$. The solution will have two constants which you need to adjust to satisfy the boundary conditions, which in this case force the trivial solution.