Why Is the Source Term Crucial in Differential Equations?

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The source term in ordinary and partial differential equations is crucial because it represents the origin of disturbances, such as waves or temperature changes. Omitting the source term simplifies the equations but can lead to oversimplification, particularly in complex scenarios with multiple sources. In cases like electrostatics, while a homogeneous equation may suffice for single point charges, the situation becomes more intricate with multiple charges, necessitating careful consideration of boundary conditions. Ultimately, understanding the source term is essential for accurately modeling and solving real-world phenomena. The complexity of the problem remains constant, regardless of the approach taken.
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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