Does the principle of superposition apply to the 1st order differential equations?

Answer 1

Yes, but only if it is linear. Here's why:

We say a function/transformation/whatever-you-wana-call-it is linear if f(cx) = c*f(x) for c is a constant and if f(x+y) = f(x) + f(y). This is indeed the case for derivatives! d/dx[2x^2] = 2*d/dx[x^2], and the additive property also holds. So if we have a y=solution to my differential equation and then w=another solution, then d/dx(y+w) = d/dx(y) + d/dx(w). If you integrate both sides, you find a true relationship (i.e., y+w = y+w), meaning that this differential equation is true always. You can see how this goes to bunkers if it were not the case that our differential equation is not linear, I'll leave that as an exercise.

Makes sense? if not, hmu!

Answer 2

The principle of superposition applies to linear differential equations regardless or order. For example, the wave equation is second-order linear partial differential equation, and therefore you can superpose the solutions.

δ²u/δt² = c²∇²u

where c is a constant. So, even though the second derivatives of u in space affect the second derivatives of u in time, if you look at the solutions (e.g. as Fourier series) the lack of cross-terms (say where u * u) As a result, even though this equation is second order (the second derivatives in time and space) it is still linear. Energy isn't going to be transferred from one dimension (e.g. x) to another (e.g. y) and it's not even going to be transferred from one frequency to another.

By contrast, even the incompressible Navier-Stokes equation is first-order in time, and second order in space:

δu/δt = u • ∇ u - ν ∇²u - ∇h

(where h is the external forcing and ν is the kinematic viscosity)

has a famous nonlinearity which leads to turbulence and preculdes the possibility of superposing solutions.

The term u • ∇ u (advection of momentum) is nonlinear because the velocity vector u multiplies the divergence of the velocity ∇ u. This means that the greater the velocity is changing in any direction, the more it will affect the velocity in every direction.

So, whether or not the principle of superposition of solutions can be applied to a partial differential equation depends on whether or not is linear in all dimensions, not its order in any dimension.