Why Is the Source Term Crucial in Differential Equations?

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SUMMARY

The source term in ordinary differential equations (ODE) and partial differential equations (PDE) is crucial for accurately modeling phenomena such as wave fields and temperature distributions. When the observation volume excludes the source, the equations become homogeneous, simplifying the problem. However, omitting the source term can lead to oversimplification, especially in complex scenarios like electrostatics with multiple point charges, where boundary conditions must be considered. Various methods exist to solve these equations, each suited for specific situations, emphasizing the importance of including the source term for comprehensive analysis.

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fisico30
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Hello everyone,

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
 
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Let's consider electrostatics with point charges.
Then you are right: solving without source term is all that is needed.
But you should realize that the domain to be considered will become more complicated if many charges are involved. And you will need to use boundary conditions around each of these charges. The simplification is a pure illusion. But there are indeed many methods to solve these problems, each with there specific advantages in specific situations.

In the end, the quantity of information to be taken into acount remains the same.
 

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