- #1
Vitani11
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- 3
Homework Statement
I have taken ODE, linear algebra, mechanics, math physics, etc. and we would always go on about how important the homogeneous equation is. To solve for the equation of motion for a harmonic oscillator (for example) we would solve for both the homogeneous and particular solution to get the general solution. I am now in PDE and this is coming up often for the heat equation. Now I would like to know what exactly it means for you to have to find the HG solution and particular solution from a more conceptual standpoint. From looking at the equations it seems to be that the HG equation represents an equation that doesn't have anything affecting it. So for a damped harmonic oscillator this thing would be dampening to 0 or for the heat equation this thing would not be losing any heat and so ∂u(x,t)/∂t = 0 (Keeping boundary conditions and initial conditions simple). I noticed that regardless of whether there is a particular solution there is always a homogeneous solution. Is that just what this means though? So when you find the general solution to an equation you find the solution corresponding to the part of the equation which is not influenced or HG (i.e. it's "natural state") and the solution to the part of the equation which is influenced by parameters or what have you. Is this correct?
Homework Equations
examples:
1. ∂u(x,t)/∂t= ∂2Φ/∂x2 where ∂2Φ/∂x2 = 0
2. d2x/dt2+2βdx/dt+ω2x = 0 for HG
and d2x/dt2+2βdx/dt+ω2x = f(t) for particular
The Attempt at a Solution
None specific