Analytical solution to Poisson’s equation 1D

Question

I need some help in finding u(x) analytically where equation and the boundary conditions are satisfied

Answer 1

Anything wrong with integrating twice?

$$u(x) = -\frac{x^2}{2} + A x + B$$

$$u(0) = 1 \implies B = 1$$ $$u(1) = 2 \implies -\frac12 + A + B = 2 \implies A=\frac{3}{2}$$