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my question is regarding the source term in ODE and PDE.

If the region where the phenomenon (wave field, temperature,...) is observed is circumscribed to a volume not containing its source, then the differential equation becomes homogeneous (no source term) and easier.

So why solve the inhomogeneous eqn ever, unless we are inside the source, since our volume of observation can always omit the source?

Clearly, a source must exist somewhere to create the dusturbance.

thanks