Question

Assuming particular solution when nonhomogenous part is constant

Answer 1

There are a number of different techniques for solving linear inhomogeneous differential equations. The simplest, which is very useful for simple right hand sides like this one, is called undetermined coefficients. Here you find the general solution to the homogeneous problem and then assume a simple form with undetermined coefficients for a particular solution. The most common case is that when the right hand side is a polynomial, you assume a solution in the form of a polynomial.

In this case, you can assume a constant solution u≡c, because if u is constant then d2udx2 is zero, and so your equation reduces to c=k