How to interpret equations with differentials?
Consider the following equation:
Geometrically, this equation represents a sphere centered at with the radius . The technical term for this is implicit surface because the relationship between , , and is defined implicitly.
The surface above can be also defined as:
This equation has the following geometric interpretation.
Consider a tangent plane touching the implicitly defined surface at some point . The differential equation states that for any point on that plane, the following must hold:
or more concisely,
All such planes turn out to be tangent planes to spheres centered at , and the initial condition allows us to select the "correct" sphere of radius .