Linearity of Harmonic Oscillator and Laplace's Equation

[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!

 

[HallsofIvy]

If w is a constant, or a function of t but not x, then yes, both equations are "exactly" linear.

If an equation is linear but not homogeneous, then the set of all solutions forms a "linear manifold". A linear manifold in, say, R³ can be thought of as a plane or line that does NOT contain the origin. If u and v are two vectors in a linear manifold, there always exists a specific vector, w, say, such that u- w and v- w are in the same subspace. u+ v is NOT necessarily in that linear manifold but (u- w)+ (v- w)+ w is. That means that every solution to a linear, non-homogeneous differential equation can be written as a solution to the corresponding homogeneous equation plus a single solution to the entire equation (the "w" in the example above).