# Linearity Question

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

## 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!