On the role of the boundary conditions in the linear combination of solutions of any linear differential equations

Question

When we have a linear partial differential equation, we can take any linear combination(superposition) of any of its solutions and it will be a solution by itself. My question is if the solutions that we are adding need to satisfy the same boundary conditions of the problem?

For example, if I am trying to solve the heat or diffusion equation with some wierd boundary conditions, can the whole solution be a linear combination of solutions with every solution satisfying a set of the original problem's boundary conditions?

Answer 1

The principle of superposition applies to linear homogeneous differential equations. In physical terms, this means that there are no external sources. For example, it applies to the equation $u_t-u_{xx}=0$, but not to the equation $u_t-u_{xx}=f(x,t)$. If $u$ and $v$ are solutions of the last equation, then $w=u+v$ satisfies the equation $w_t-w_{xx}=2\,f(x,t)$.

The same applies to the boundary conditions. If they are homogeneous, like $u(a,t)=0$, or $u_x(a,t)=0$, then the principle of superposition holds.