Is Linearity Always Exact in Differential Equations?

[bon]

Homework Statement

Ok so basically I have two differential equations:

1) x'' + wx' + w^2 x = sin5t

2) del squared u = 0

The first is obviously just the equation for the driven harmonic oscillator. The second is Laplace's equation.

The question asks: In both cases, is the linearity believed to be exact, or the result of an approximation (if so, say what it is). In the case of a second order PDE what are the properties of solutions that follow when it is (a) linear (b) linear and homogeneous

Homework Equations

The Attempt at a Solution

Ok so

Is the linearity exact in both cases? I am pretty sure it is in the ODE case, right? Is it also in the PDE case (i.e. 2) ? ) when is linearity not exact?

For the second question:

I guess that when the PDE is homogeneous and linear then linear sums of solutions are also solutions, but what about when it is just linear?

Thanks!

 

[bon]

anyone?

 

[sigmaro]

actually i advise you to move your question to advanced physics or mathematics forums, there especially at mathematics part maybe experts can help.
i haven't legally studied these stuff much, but i once heard definition of linearity is f(kx)=kf(x)
so f(0)=0 more, it must be like f(x)=ax
both these equations are combinations of exp(aix), exp(-bix), exp(cx) and exp(-dx)
and they definitely don't have to be 0 at x=0
i am quite surprised, i was surprised when i had first heard but i trust mathematics :)
btw i used
http://en.wikipedia.org/wiki/Nonlinear_system#Definition
http://en.wikipedia.org/wiki/Linear_map#Definition_and_first_consequences

 

[Redbelly98]

Just use the definition of linearity:

1. If u is a solution, then k*u must be a solution, where k is a constant.
2. If u1 and u2 are solutions, then ____(?) must be a solution.